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Teaching students about ma rainey: the mother of the blues, teaching students about bayer leverkusen: an introduction to the world of german football, teaching students about morphological awareness: enhancing literacy skills, teaching students about dave chappelle: a guide for educators, teaching students about the ionian sea: an enriching experience, teaching students about the santa clarita school shooting: a guide for educators, teaching students about the toyota previa: an innovative approach to automotive education, teaching students about super hero girls: empowering the next generation, teaching students about the iron curtain: a comprehensive guide, 10 apps to help students with their science homework.

Science can be hard. Science homework can be harder, especially when the teacher isn’t there to help. Many parents now are struggling to help students with their STEM and Common Core aligned science homework, no matter what specific scientific branch it is.

Rather than spend money on a tutor, surf the Internet for hours, or try to squeeze a few minutes out of a teacher’s free time, students can, instead, use these ten apps to help themselves with their homework. This way, students can take charge of their own learning and homework skills.

#1 The Chemical Touch

This amazing resource will help students ace Chemistry. A full periodic table is given on the app. Once a student touches a specific element, they can study it more in depth such as its atomic mass and properties. It also has an amino acid chart with in-depth information. Each element and amino acid also have a link to a Wikipedia article for more information.

#2 Frog Dissection

One of the most stereotypical science experiments, frog dissecting, can now be done virtually. For students who get queasy or who would prefer to avoid this experiment due to religious reasons, this app is a perfect alternative for them. Featuring step-by-step instructions, there is plenty of in-depth information on each of the frog’s organs including anatomical comparisons to human organs.

#3 Kahn Academy

With over 40,000 interactive Common Core aligned practice questions and over 10,000 videos and explanations in math, science, economics, history, and more, Kahn Academy is the perfect app for students of all ages to study. Kahn Academy is free of charge and is the perfect study resource and tutoring app for students who are struggling in science (as well as other subjects).

#4 Stephen Hawking’s Snapshots of the Universe

Based on the writings and work of Stephen Hawking, one of the world’s most renowned scientists, this app is a wonderful interactive source of information for space science. This app includes 10 interactive experiments and video segments to help students study our universe.

#5 NASA Visualization Explorer

Perfect for expanding upon concepts learned in class or research for projects, this NASA app helps students explore our universe even more. With articles and visuals, students can discover more about the earth, the solar system, and beyond in depth. NASA adds new stories every week to keep the app and its information up to date.

#6 Project Noah

Project Noah “is a tool to explore and document wildlife and a platform to harness the power of citizen scientists everywhere.” This app is a great way to get students involved in biology and ecology at home with interactive homework assignments. Rather than studying animals and plants from a textbook, students can go out in nature and take pictures of their assigned plant or animal. They can even earn “badges” which also makes this a great app for the Boy Scouts.

Students studying for the SAT, chemistry, or physics can get extra help and explanations with this app. Employing experts in the STEM standards, students can upload a snapshot of their homework question and get one-on-one help to understand the problem. Students get a free 10-minute session for each problem. It even includes SAT prep questions and practice tests.

#8 Anatomy 4D

For high school and college students, Anatomy 4D is a great resource for studying anatomy. Students can study specific parts of the body or the entire body. Views can be switched from male body to female body. Views are in 3D to give a more life-like experience to students using the app. Information is also available for students to read as they study specific organs or systems.

#9 Physics Calculator

High schoolers who need extra help with their physics homework can benefit from this app. With fill-in-the-blank equations for kinematics, motion, energy, power, gravity, temperature, thermodynamics, and many more physic properties, this app is a quick solving tool for those hard to answer physics problems.

#10 iCell App

Studying the cell became easier with this 3-D cell app. Students can view the cell structures and dynamics for plants, animals, and bacteria. There are even different levels of information for students varying in age and grade level. Students can zoom in on different parts of the cell and compare cells with each other. It even includes color-coded DNA and more in-depth information about cell size and scale, skin, and much more.

This list is not all-inclusive—there are many more apps in all the branches of science available as resources and interactive games to help students study and learn more about science. By relying on these apps, students can learn more about science while doing their homework and develop a love of learning on their own time.

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10 of the best science apps for Android

Turn your Android phone into an all-knowing guru with our top ten favourite science apps.

If you find yourself tinkering around the same old games on your smartphone, maybe it's time to put some of that processing power to good use and give your brain a bit of a workout. Here are some of the best android apps for anyone interested in a little science, including a spot of VR, some sumptuous nature, and yes, a calculator (we promise it's a good one).

• Science podcasts: 10 excellent BBC podcasts to boost your brain
• 26 of the best AR and VR apps and games for iOS and Android

Get your head around one of the toughest problems in biology with this free, Tetris -style game, developed by scientists at two London universities.

BioBlox is inspired by the problem of protein docking – working out how exactly molecules such as medicines and vitamins bind onto complex proteins in the body.

There’s a 3D version of the game, too, which the creators hope to use in citizen science projects to solve real-world protein docking problems.

• Goldsmiths/Imperial College London , free

acceleratAR

This clever app allows you to create your very own (virtual) particle accelerator, using nothing more than a smartphone and some paper cubes.

Position the cubes on a flat surface (each cube represents a different component of the accelerator, such as a particle source and an electric field) and then bring it to life on-screen using the accompanying app.

• Cockcroft Institute / University of Liverpool , free

Collins Bird Guide

The ultimate companion for any serious birder, this digital version of Collins’ bird bible covers over 700 European species, with 3,500+ illustrations helping you to tell your ruddy shelduck from your red-crested pochard.

Once you’ve identified the bird, learn about its habitatand range, and record your sighting using the listing tool.

Bill Oddie, eat your heart out.

• NatureGuides , £12.99

For anyone who’s ever wanted to learn to draw, this app provides a clever way. SketchAR lets you choose an imageand then hold your phone or tablet over a blank piece of paper to see that image appear on-screen. You can then simply trace what you see, replicating the image line by line to create something that even Picasso would be proud of.

• sketchar.tech , free(onlyavailable on smartphones with Tango)

Pocket Code

This nifty introduction to programming lets you create, play, and share your own games, animations, and music videos using a visual, ‘Lego-style’ interface.

Visit pocketcode.org to download and remix others’ creations, ranging from Space Invaders -style shooters to a visual proof of Pythagoras's theorem.

• Catrobat , free

All-in-One Calculator

We love a good calculator here at BBC Science Focus , and this lovely looking Android app promises to fulfil all our mathematical needs.

It features over 50 different calculators and unit converters, ranging from algebra and geometry to health and finance.

MindBox Apps ,free with in-app purchases

Thinkrolls: Kings & Queens

Tear through a land of treacherous castles in this follow-up to the award-winning Thinkrolls 2 .

There are 228 puzzles for ages 5+, designed to sharpen their problem-solving skills, exercise their memory and introduce them to forces and basic mechanics.

Ghosts, dragons and physics… what’s not to like?

• Avokiddo , £2.99

Attenborough’s Story of Life

To celebrate the great broadcaster’s 90 th birthday, this app brings together over 1,000 (count ‘em!) of Sir David Attenborough’s finest TV moments.

There are highlights from more than 40 programmes, including Planet Earth , Blue Planet , and Africa , as well as curated collections of films and hidden extras recorded especially for the app.

• BBC Worldwide , free

Notes on Blindness VR

In the 1980s, the theologian John Hull lost his sight. He began keeping an audio diary, a document which formsthe basis of the critically acclaimed film Notes on Blindness , released earlier this year. This accompanying VR experience beautifully recreates memories and moments from John’s audio diary, immersing the viewer in a ‘world beyond sight’.

• ARTE Experience , free

Cosmolander – Missions in the Solar System

Inspire the astronomers of the future with this introductory guide to our Solar System. Aimed at kids aged six and up, they’ll learn about the science and history of each planet before embarking on missions to test their knowledge, from repairing a Martian rover to collecting diamonds on Neptune.

• Marshmallow Games , £2.69

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Train the skills that matter to you most.

Memory. Processing Speed. Problem Solving. Lumosity targets these cognitive skills and more.

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Lumosity takes tasks from the lab and turns them into fun games. We interpret your scores to offer actionable feedback and rich insights into your cognition.

Daily exercise for your mind

Work out with a fresh set of games each day to keep you challenged. Detailed progress tracking helps maintain your brain training habit.

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No matter your age or skill level, Lumosity knows that all brains are different, and our program adapts to your unique strengths and weaknesses.

Science. That feels like games.

Our scientists take tasks from the lab and adapt them into easy-to-learn brain games., 14 years, 100 million members.

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Researching the efficacy of Lumosity

What we did

Lumos Labs conducted a randomized study of Lumosity brain training and published the results in a peer-reviewed research journal.

In it, half of the 4,715 participants who completed the study trained five days per week, for fifteen minutes each day on Lumosity while the other half did online crossword puzzles as an active control.

What we found

After 10 weeks, Lumosity users improved more than the control group on our assessments of working memory, short term memory, processing speed, problem solving, fluid reasoning, and overall cognitive function.

These results are promising, but more research is needed to determine the connection between improved assessment scores and everyday tasks in participants' lives.

Next questions

Future research should address the risk of inadvertent experimenter bias and the risk of attrition bias in this study, as both the Lumosity and crossword groups had approximately 50% attrition rate. As with all scientific research, there is also a risk of publication bias.

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Build a creative practice by experimenting with music, art, writing and more. download our newest app, figment, to jumpstart your creativity with new daily activities., introducing lumosity mind, lumosity mind includes mindfulness sessions on the topics of relaxation, focus, and sleep—designed by the experts at lumosity., start your free training program.

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STEM Apps for Higher-Order Thinking

These seriously fun apps help students develop and apply a conceptual understanding of science, technology, engineering, and math by solving puzzles, designing solutions to novel problems, and creating new inventions. It's important for students of all ages to engage in this type of critical thinking to prepare for college and career. Check out our list of tools that get students coding robots, engineering new creations, playing with games and simulations, and gathering and analyzing scientific data. 

Engineering and Inventions

Crazy gears.

Highly applicable STEM-themed free play; easy appeal for wide audience

Bottom Line : A clear, simple, and fun introduction to science and engineering topics that may otherwise feel unapproachable for some kids.

Hero Elementary

Cute science and engineering activities can fill gaps, feed curiosity

Bottom Line : These games can be a great supplement to a science class, though each individual activity works best when played in a single session.

Simple Machines by Tinybop

Excellent elementary NGSS engineering practices embedded in fun games

Bottom Line : Fun gameplay lets kids explore important physics concepts in a solid standards-aligned context.

The Robot Factory by Tinybop

Improve designs, collect favorites in robot creation sandbox

Bottom Line : A well-made and enjoyable STEM app that leverages what's great about play -- experimentation -- to spark engineering interest.

Inventioneers Full Version

Budding engineers create design solutions with fun physics puzzler

Bottom Line : A fun, engaging way to boost critical-thinking and problem-solving skills while learning about important science and engineering practices.

Gizmos & Gadgets

Top-notch tool empowers kids to invent, build, and control wirelessly

Bottom Line : Easy-to-use, versatile electronic invention set that works wirelessly with your hand-held device.

Kids use block code and wireless blocks to program, design, and create

Bottom Line : With some perseverance, it's a great fit for PBL and STEAM, helping students learn programming skills and engineering concepts through invention.

Stump students with innovative train-routing puzzler

Bottom Line : Superb combination of logic, real-world learning, and train play.

Break down concepts in a new way with AR models

Bottom Line : These 3D STEM models will certainly engage students when added to lessons, but usefulness is limited by a lack of creation options.

World of Goo

Stellar physics builder with wiggly, animated goo balls

Bottom Line : A fascinating fantasy world and constant puzzle challenge pull kids into learning about building structures and geometry.

Coding and Robotics

Codeable crafts.

Animate stories with accessible drawing tools and coding blocks

Bottom Line : Kids get a simple peek at computer programming principles by animating stories using their own creations.

Blockly for Dash & Dot Robots

Adorable robots teach hands-on coding skills

Bottom Line : You can program robots with code blocks, with plenty of creative options and lots of fun.

Evo by Ozobot

Versatile mini robot tailor-made for teaching coding

Bottom Line : Evo encourages students to be creative, emphasizing that coding is a tool in your pocket, not the end game.

Cool robots and creative play make programming irresistible

Bottom Line : Hub for Sphero robots draws crafty kids who love making into coding, and it empowers them to be collaborators.

Swift Playgrounds

Brilliantly designed iOS coding app great for newbies or pros

Bottom Line : If you have iPads or Macs, this is an inspired choice for learning how to create and tweak code.

Challenging game helps kids learn to think like programmers

Bottom Line : Cargo-Bot is a simple concept that packs a challenging punch and teaches valuable programming skills.

Science and Math

Bugs and bubbles.

Fascinating graphics engage as kids practice pre-reading, STEM skills

Bottom Line : Great bang for the buck with all the content and stages included.

Nico & Nor Coconut Star

Early education force and motion app ramps up student learning

Bottom Line : Coconut Star, along with its companion app, are great additions to any pre-K–1 physical science unit.

Osmo Tangram

Fun hybrid puzzles with excellent classroom supports

Bottom Line : Classic puzzler combines toys with digital play to engage and teach all ages.

Math Makers

Charming leveled puzzler slyly mixes fun with logic, essential math

Bottom Line : These puzzles are a really fun approach to developing an intuitive sense of important math concepts, particularly for kids who feel that math is boring or inaccessible.

Shifu Plugo

Augmented reality app has appealing STEM puzzles, room to grow

Bottom Line : Shifu Plugo is a good AR product that, with continued development, could be great for collaborative learning.

Playground Physics

Fun video app lets students track motion to explore physics concepts

Bottom Line : This is a cheap way to get students out of their chairs to make physics real and tangible, empowering learners and encouraging curiosity.

Light & Color by Tinybop

Open-ended scientific exploration shines best with offline scaffolding

Bottom Line : Free-play science-based exploration works best for students who love to discover things on their own with little guidance; others will need scaffolding.

Turn kids into budding nanoscientists who explore the tiniest things

Bottom Line : Scientist-reviewed videos and activities relate nanoscience to everyday life.

Approximate photos of real-world objects with virtual 3-D shapes

Bottom Line : A quick way to make solids more tangible, especially when paired with good lesson plans that get students working directly with formulas.

Lab4Physics

Bring physics to life using lab sensors built right into your smartphone

Bottom Line : Use your smartphone to gather data and study speed, gravity, waves, and other physics concepts.

The PocketLab

Gather and analyze live data anywhere

Bottom Line : A science sensor that pairs with your device to gather and analyze data is perfect for inquiry-based learning.

Arduino Science Journal

Turn your phone into a lab sensor to collect and analyze data

Bottom Line : Science Journal does a good job recording real-time motion, sound, and light data using phones.

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Best Science Apps for Android and iOS

Smartphones are becoming powerful day by day. Other than using all the processing power to play games or run some general category apps, you can use the power to satisfy your curiosity or become a citizen scientist solving complex problems. If you are a student, researcher, or hobbyist in the scientific field, here are some of the best science apps you can try.

Best Science Apps

1. science journal.

Price: Free

Science Journal is a digital notebook geared towards researchers, students, and hobbyists. The app provides an easy way to quickly add your observations with notes and photos in a place with just a few taps. Moreover, since the app is developed by Google, you can also sync your notes and journals between devices using Google Drive.

Science Journal is quite different from other journal apps. It has built-in tools that make use of your phone’s sensors like accelerometer, gyroscope, ambient light sensor, etc. Using these sensors, you can measure a wide variety of things like light, sound, air pressure, acceleration, motion, angles, etc. Additionally, you can also connect external sensors via Bluetooth.

Additionally, the app allows you to automate certain tasks like the recording of data, the ability to export sensor data, use the built-in camera app to scan and take photos, etc. All in all, if you are looking for an app to keep track of your findings, observations, and research then Science Journal is a pretty good app to try.

Download Science Journal ( Android | iOS )

2. WolframAlpha

Price: $2.99

The science app list won’t be complete without adding WolframAlpha. Using WolframAlpha, you can compute and analyze a wide variety of things like mathematics, physics, chemistry, data analysis, statistics, materials, life sciences, etc. No matter your field, WolframAlpha probably covers that domain and can help you with whatever computational needs you have.

As you can guess, WolframAlpha makes your life easier when you need precise and quick answers to complex questions. Simply put, if you are someone who deals with a lot of computational questions then WolframAlpha is a must-have app.

Download WolframAlpha ( Android | iOS )

3. Star Chart

Price: Freemium

Start Chart is one of my favorite apps. With Start Chart on your device, you get a clear and a whole new look at the stars and the universe. Personally, I have a hobby of looking at distant planets. However, I am terrible with directions and remembering names. So, I use the Start Chart app to quickly find the position of a planet or a star and then use the hobby telescope to take a quick look. Star Chart does all this by using your phone’s GPS and built-in 3D universe.

Of course, I’m no professional. However, if you are like me or a professional, Star Chart is greatly helpful in finding known planets and stars among other things.

Download StartChart ( Android | iOS )

Of course, NASA is one of those apps that should be included in every science apps list. It’s the official app and gives you instant access to the latest space news, over thousands of real and awesome photos, satellite tracking 2D and 3D earth models, mission information, NASA TV, live high definition earth viewing from International Space Station, etc.

If you are into space and astronomy then NASA is a must-have app.

Download NASA ( Android | iOS )

5. Calculator N+

Your phone’s inbuilt calculator might suffice normal day-to-day calculations but when it comes to scientific problems, it’s hardly of any use. For that, you can use Calculator N+. It can perform manual calculations of complex equations and problems. If needed, it can also draw graphs for you.

Do keep in mind that unlike WolframAlpha which can deal with a whole lot of different domains, Calculator N+ only supports Mathematics. Additionally, it will not show the steps to solve an equation, only show the final result.

If you are looking for an easy to use scientific calculator that is filled with features then Calculator N+ is for you.

Download Calculator N+ ( Android )

6. Microsoft Maths Solver

Microsoft Maths Solver is another scientific calculator app. Compared to Calculator N+, there is 2 major difference where the app stands out. First, it can scan equations from a piece of paper, and second, it also displays the steps and formula taken to solve the equation. You can also switch between different methods available or view of a similar video demonstration.

Similar to the Calculator N+ app, it also shows the graph calculation. The app also has a web version so you can access it on your desktop as well. If you are a physics, math student, or researcher, Math Solver can be very helpful in a pinch.

Download Microsoft Maths Solver ( Android | iOS )

7. Chemistry & Physics simulations

If you are chemistry, physics student, or a researcher then this app is a must-have. The app is mainly designed with educators and students in mind. hence, the app is super simple to use and follows an interactive game-like environment to show chemistry and physics simulations. Of course, being a simulator, you can mess with a lot of things as and when needed and see the results almost instantaneously.

Download Chemistry & Physics simulations ( Android )

Read: Best Learn to Code Apps (Android & iOS)

8. Periodic Table 2020

Price: Free (has ads)

Back in school, I had the huge Mendeleev’s periodic table stuck on my wall. It was a huge utility for chemistry and this app solves the problem. The app is regularly updated and consists of Mendeleev’s periodic table. You can click on any element on the table for additional information like atomic weight, atomic number, valency, etc.

Download Periodic Table 2020 ( Android )

9. Pocket Code

Now, coding is not necessarily a scientific app. However, when it comes to science, coding is very important. Learning to code allows you to think logically and even improves problem-solving skills. If you’ve never learned a programming language then Pocket Code can help you get started. Pocket Code uses the Lego-style programming language where you place blocks in a specific manner to create your own applications and games.

So, if you are new to programming, do give Pocket Code a try and see how it works for you.

Download Pocket Code ( Android | iOS )

Read: Best Math Game Apps for Android and iOS

10. CamScanner

When it comes to scientific stuff, there are a ton of papers to deal with. The good thing is, you can easily digitize them so that you not only store them more efficiently but also make it easy to read. CamScanner is a very simple yet hugely popular app that lets you scan documents and create PDFs. Once a PDF is created, you can export it as is or in any other supported format like JPG and PNG.

Download CamScanner ( Android |  iOS )

11. Duolingo

Duolingo is a popular app to learn foreign languages for a beginner. The app will help you learn French, Spanish, German, along with 20 other different languages. The cool thing about Duolingo is you don’t need to know English to use the app, it works with your native language as well. While no app can help you learn a foreign language completely, Duolingo does help you to get proficient.

Download Duolingo ( Android | iOS )

12. Khan Academy

While you can always google or watch lessons on YouTube, it’s better to have an app with a proper curriculum. Khan Academy is a free learning app and offers courses for students from kindergarten to graduation. These courses cover diverse subjects like maths, science, history, humanity, arts, economics, computing, etc. The courses are well categorized and you get interactive tests as well. Additionally, the app also features partner content from sources like The British Museum, American Museum of Natural History, etc.

Download Khan Academy ( Android | iOS )

Closing Words

That’s it for now. I also spend a lot of time on subreddit such as AskHistorians, dataisbeautiful, tech, TIL, WorldNews, etc. If you have subscribed to the right subreddit, it can boost your information to a great extent. In case you think I missed any of your favorite apps then comment below and let me know.

Also Read: Top 7 Homework Planner Apps for Students

Vamsi Krishna

Vamsi is a tech geek who enjoys writing how-to guides and tinkering with computers and software in general. He especially loves using Windows and writing about it. You can check out his Windows guides at WindowsLoop .

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25 Critical Thinking Apps For Extended Student Learning

What Are The Best Critical Thinking Apps For Extended Student Learning?

by TeachThought Staff

This post may contain affiliate links. You can read more about affiliate policy here.

Critical thinking apps aren’t the sweet spot of mobile app design.

Most of the more popular apps are focused on social media, productivity, or what always seem to be mindless games. The fact that most apps  don’t  force users to think critically doesn’t mean that it isn’t possible, however. The lack of cognitively demanding apps is more of an indictment of the users than the technology itself.

There are many apps that do promote critical thinking–and often extended critical thinking and learning at that. These aren’t clinical ‘critical thinking building’ programs either, but rather often enjoyable exercises in strategy, tactics, and problem-solving thought. So below, we’ve collected 25 of these critical thinking apps. Most are for grades 8-12, but several are for students as young as kindergarten.

See also What Is Critical Thinking?

Helping students think critically is less a matter of direct instruction, and more a matter of providing them with access to dynamic tools with unique and playful barriers or goals that require the best strategy they can muster, while providing instant feedback to the player as they go. Though they’re not learning specific academic content in most of these apps, they are practicing extended thinking over a period of time that encourages them to look at digital circumstances with a critical eye.

1. Google Earth

What is Google Earth?

Google Earth is a virtual globe and mapping software that allows users to explore the Earth’s surface and navigate through detailed satellite imagery, aerial photographs, and 3D models of various locations. It provides a comprehensive view of our planet, enabling users to zoom in on specific areas, search for specific addresses or landmarks, and obtain information about different geographical features.

Google Earth also offers features such as Street View, which provides immersive 360-degree street-level imagery, and Voyager, which offers curated interactive guided tours of fascinating places around the world. Whether used for educational purposes, travel planning, or simple curiosity, Google Earth serves as a powerful tool that brings the world to users’ fingertips, promoting exploration and a deeper understanding of our planet.

2. Blinkist

Blinkist is a digital platform that provides concise summaries of non-fiction books in the form of bite-sized content, allowing users to quickly grasp the key insights and main ideas of a wide range of books. With a vast library covering various genres such as self-help, business, science, and more, Blinkist seeks aims to make learning more accessible and time-efficient for people who want to stay informed and expand their knowledge but may not have the time to read entire books.

By condensing the main concepts, Blinkist allows users to get the essence of a book in just a few minutes, making it a popular choice for busy individuals seeking personal and professional growth.

3.  Elevate

Grade Levels:  8-12+

Developer Description

Elevate is another app that focuses on improving cognitive skills, including critical thinking. It offers a variety of games and challenges that target different areas, such as memory, comprehension, processing speed, and analytical thinking. Elevate also provides personalized training programs based on your goals and tracks your performance.

4.  Minecraft

Grade Levels:  K-12+

“Explore randomly generated worlds and build amazing things from the simplest of homes to the grandest of castles. Play in creative mode with unlimited resources or mine deep into the world in survival mode, crafting weapons and armor to fend off the dangerous mobs.”

5. Critical Thinking

6.  Civilization VI  (Note: For Mac, not iPad) (Note: Editor’s Choice award from TeachThought)

Grade Levels:  6-12+

“Originally created by legendary game designer Sid Meier, Civilization is a turn-based strategy game in which you attempt to build an empire to stand the test of time. Become Ruler of the World by establishing and leading a civilization from the Stone Age to the Information Age. Wage war, conduct diplomacy, advance your culture, and go head-to-head with history’s greatest leaders as you attempt to build the greatest civilization the world has ever known.”

7.  Brilliant

Developer Description8. Principles In Action

9.  Shakespeare In Bits: Romeo & Juliet

Grade Levels:  7-12+

“Understand and appreciate the linguistic richness of Shakespeare’s plays with the original play text, broken into easily digested ‘bits’. In-line translation helps you to understand difficult words and phrases within the context of the original play text.

Each title includes hours of high-resolution animation with a professional audio soundtrack, helping you to decode and comprehend Shakespeare’s fertile prose quickly and efficiently. The text, presented side-by-side with animated action, is easy to follow with portions of text being highlighted as they are spoken in the play. The integrated combination of text, audio and graphics is designed to facilitate learners of varying abilities and learning styles.”

10.  Literary Analysis Guide

“Literary Analysis Guide is a reference tool for students of literature, rhetoric, and writing. The elements of literature are arranged around three clickable interface wheels. Touch any term on the wheels for a detailed definition, examples from literature, questions to ask, and an example essay paragraph that analyzes that term. Students can also reference an illustrated version of each analysis paragraph which highlights the topic sentence, textual evidence, and analysis portions of text.”

11. SPEEDSim: Spatial Population Ecological and Epidemiological Dynamics Simulator

Grade Levels:  9-12+

“Spatial Population Ecological and Epidemiological Dynamics Simulator (SPEED Sim) is a tool that enables hands-on interactive exploration of the spatial dynamics of various computational models in population ecology and epidemiology. The app also provides several cellular automaton models as an introduction to these kinds of simulations.”

12. MasterClass

What Is MasterClass?

MasterClass is an online learning platform that offers in-depth video courses taught by renowned experts in various fields. It provides individuals with the opportunity to learn from some of the world’s most accomplished professionals, including celebrated actors, musicians, writers, athletes, chefs, and more. Each course consists of pre-recorded video lessons, typically divided into several chapters, where instructors share their knowledge, experiences, and techniques.

The classes seek to be well-produced, combining instructional content with personal anecdotes and demonstrations to create an immersive learning experience. With a diverse range of subjects available, MasterClass empowers learners to explore their passions, acquire new skills, and gain insights from industry icons.

13. Audible

Animate Me! 3D Animation For Kids 

Grade Levels:  3-12+

“Create 3D animations of cartoons in a simple and intuitive manner on your tablet or phone. If you’ve ever wondered how they bring characters to life in animation movies like Frozen or Toy Story, this app is for you. Kids can learn to make 3D animations that showcase their creativity and storytelling abilities.”

14.  Opposing Views

Grade Levels:  10-12+

“At Opposing Views, we report the news and let you debate it. We deliver the day’s most pressing, important and unbiased news. Instantly access news that will make you think, laugh…”

What is TED?

TED is an app that provides access to a vast library of TED Talks and presentations on various topics. Listening to TED Talks can stimulate critical thinking by exposing you to different perspectives, ideas, and arguments. The app allows you to explore talks on subjects like science, technology, psychology, and philosophy, encouraging you to think critically and engage with thought-provoking content.

16.  Neuriva Brain Gym

“Shake and tilt sugar molecules to set off chemical reactions and earn points! This game features a cast of protein characters based on real structures found inside our cells. Explore the structure and function of ten of the tiny proteins that break down sugar in the cell. In each level, tilt different sugars to their matching proteins, setting off chemical reactions and avoiding obstacles.”

17. TinkerBox HD – Autodesk Inc.

Grade Levels:

“If you can imagine it, you can build it with TinkerBox. TinkerBox is a fun, free-to-play physics puzzle game. While it is full of interesting science facts and teaches basic engineering concepts, TinkerBox is more than just educational! Take the tools in your hands to explore your creativity and imagination with Invent mode. Build outrageous machines, share them with your friends, or download popular inventions online. Stretch your brain through the devious Puzzle mode. Get immersed in physics-based puzzles and mechanical concepts, where your only hope for success is your creative problem solving.”

18.  SimCity BuildIt – Electronic Arts

“Solve real life challenges like traffic, fires, and pollution. Provide services like power plants and police departments to meet your citizens’ needs, and boost your population with parks and education. Keep traffic moving with grand avenues and streetcars. Complete fun challenges to shape your society.”

19. Artificial Life

“Oddly hypnotic and relaxing, Artificial Life is a simulation of the growth and evolution of microorganisms – basically artificial life forms. Initially, each organism is given a random set of rules governing its behavior. If the rules end up being useful for survival, the organism may live long enough to mate with another organism and pass its genetic code on to some offspring.

Over time, the organisms with the most useful behaviors will pass on their genetic code, leading to interesting patterns of behavior unfolding before your eyes – flocking behaviors, solitary predatory tactics, scavenging, and hiding are some examples – and all this behavior is evolving from total randomness.”

20.  Block Craft 3D : City Building Simulator

Basic Description

Like Minecraft, with a greater focus on crafting.

20.  The Sims

Build a life, make decisions, design homes, prioritize relationships, maintain personal health, focus on actions and decisions that promote health, happiness, and well-being.

For other “micro-life simulations,” consider:

21. Chaos Lab

22. LifeSim

23.  SimpleLife 

24.  Artificial Life

25.  Game Of Life

Full Affiliate Disclosure

External links on our website may be affiliate links that could result in us receiving compensation (payment) when you click on the link. For example, we may receive pay per click revenue or commission on sales of products. This varies from link to link–the vast majority of links provide us with no compensation, and any compensation we might receive is low (e.g., usually 6% or less). This helps us to, among other things, cover cost of content product and consistently deliver content.

Regardless of affiliate links, we try very hard not to share products or services to you that won’t help you as an educator in some way, shape, or form. We turn down dozens of posts, ad campaigns, and other for-cost content every year due to a lack of alignment between said content, and what we perceive your needs as a progressive educator to be. If you ever have any questions or comments about this practice, email Terry Heick directly.

About The Author

Teachthought staff.

Summer is a great time to explore cool problems and keep your skills sharp!   Schedule a class today !

Train With the World's Top Math Minds

Perfect for grades 5–12. Since 1993, our students have mastered the critical thinking and problem solving skills to excel in prestigious competitions, universities, and careers.

Since 2015, all USA International Math Olympiad teammates have been AoPS students

Art of problem solving trains students to approach new challenges by breaking problems down into familiar parts., critical thinking.

Students focus on advanced problem solving and conceptual understanding—not rote memorization

LIVE COLLABORATION

Discuss new problems with brilliant peers and mentors from the world's largest problem solving community.

Face new challenges every day and build the confidence to approach future problems without fear.

On Track to Prestigious Universities

Our students earn admission to the most prestigious universities in the nation, including:

UC BERKELEY

Many of our students who are mathematically inclined have found Art of Problem Solving (AoPS) an indispensable resource. – MIT Admissions Office

Olympiad Winners Train the Next Generation

AoPS curriculum is designed and taught by former math and science Olympians.

VIEW COURSES

Staff Spotlight

Richard Rusczyk

Founder and CEO USAMO Winner Perfect AIME

David Patrick

Principal Math Curriculum Editor USAMO Winner Putnam Top 10

Principal Math Olympiad Curriculum   Developer Most Highly Decorated IMO Winner

Senior Math Curriculum Developer Winner of Canadian Mathematical   Olympiad 3-time Director of Canadian IMO Team

Ashley Ahlin

Online Math Instructor MATHCOUNTS Problem Writer US Chem Olympiad Team Member

Mark Eichenlaub

Physics Curriculum Team Lead Coach of the US IPhO Team

AoPS Alumni Success

In the last 9 years, 53 USA International Math Olympiad team members have medaled and have taken over 300 AoPS Online courses.

1 Million

problem solvers discuss and solve challenges together on AoPS Online—one of the largest online math communities in the world.

AoPS online school helps thousands of students each year develop tools needed for success in top-tier colleges and in prestigious math competitions. – Mathematical Association of America (MAA)

Ready to join our community? Pick a class that best fits your interests and schedule.

Challenge your student to reach their fullest academic potential.

Learn from the leaders in advanced math, alongside outstanding classmates, with interactive online math for students grades 5–12.

Online Math Courses

Join our live, interactive math classes with expert instructors and talented peers.

Gain the critical problem solving strategies you need for college and career success.

More Than Online Math: A Creative Problem Solving Community

Welcome to the world of AoPS Online: A world filled with challenging problems that require creativity and persistence to solve. A world where students learn from impressive mentors and role model instructors. A world filled with inspiring peers who share their love of learning.

Our advanced math curriculum stretches students to their fullest potential. And the connections students make in the AoPS classroom and online community last a lifetime.

Every class I've taken with AoPS has been fantastic. Everyone does a great job helping students succeed while enjoying the process of learning math. I love that the students themselves are always open to helping each other out.

– Matthew

I've developed a lot more confidence in math since taking AoPS Introduction to Geometry. In the process of doing harder problems than I ever knew I could, I've also begun to see geometry as an art. I'm enjoying math more and more!

– AoPS Online Student

AoPS courses are a wonderful balance of challenge and guidance, constantly stretching students and inviting them to wrestle with problems at their own pace until they reach epiphanies. Bravo! I wish AoPS had been around when I was in high school.

– AoPS Online Parent

My instructor was great. She was always willing to offer advice and knew just how many hints to give to keep problems doable but still satisfying.

My son is eating your classes up. He's always loved math, but cannot get enough of it now. He is pumped for each Wednesday. Way to provide the best educational product this Dad has found anywhere!

– John C.

The combination of videos, textbooks, online problems and book problems has been fantastic for our boys. I love that they can do this at their own pace, and how motivated they feel now that they're in charge of their own learning.

All members of the 2015–2023 US International Math Olympiad teams are AoPS alumni.

Online Math Course Categories

AoPS Online courses take place weekly in our interactive online classroom. Expert instructors lead cohorts through engaging and rigorous challenges, and students work with peers to creatively problem solve together.

Introductory Math

Grades 5–10+ Prealgebra to Geometry

Intermediate Math

Grades 8–12+ Intermediate Algebra to Precalculus

Advanced Math

Grades 9–12+ Calculus to Olympiad Geometry

WOOT Programs

Grades 9–12 Worldwide Online Olympiad Training in Math, Physics, and Chemistry

Contest Prep Math

Grades 6–12 Preparation for Math Competitions

Computer Science

Grades 6–12+ Programming Classes

AoPS offers our advanced math books in both online and paperback format. Our comprehensive math curriculum covers prealgebra all the way through contest prep math.

Complete introductory math curriculum for students grades 5–10 that covers the standard Prealgebra/Algebra/Geometry series in depth as well as important discrete math topics.

Intermediate & Advanced Math

Challenging math textbooks that prepare students grades 8–12 for advanced mathematical subjects in high school, college, and careers in STEM fields.

For more than 25 years, math competition winners have relied on AoPS problem-solving guides for contest prep. Suitable for grades 6–12, and written by medal winners in MATHCOUNTS, AMC, AIME, and USAMO.

Other AoPS Programs

Art of Problem Solving offers two other multifaceted programs. Beast Academy is our comic-based online math curriculum for students ages 6–13. And AoPS Academy brings our methodology to students grades 2–12 through small, in-person classes at local campuses.

Through our three programs, AoPS offers the most comprehensive honors math pathway in the world. View a full comparison of our program offerings.

Free Student Resources

Art of Problem Solving offers a wide variety of free resources for avid problem solvers, including hundreds of videos and interactive tools like Alcumus, our popular adaptive learning system.

Alcumus is our free adaptive learning system. It offers students a customized learning experience, adjusting to their performance to deliver problems that will challenge them appropriately. Alcumus is aligned to our Introductory Math, Intermediate Algebra, and Precalculus online courses and textbooks.

Our video library offers hundreds of videos featuring AoPS founder Richard Rusczyk. Many of the videos are aligned to our Prealgebra, Introduction to Algebra, and Introduction to Counting & Probability books and classes.

MATHCOUNTS Trainer

The MATHCOUNTS Trainer has thousands of problems from previous School, Chapter, State, and National MATHCOUNTS competitions. Full solutions are provided for every problem so aspiring MATHCOUNTS mathletes can learn how to approach even the toughest questions.

For the Win!

Compete with middle school students from around the world with For the Win! (FTW), inspired by the MATHCOUNTS Countdown Round.

5 apps to help students with science problems

Most students have turned online to clear their study doubts. if you are looking for websites to help solve your science subject doubts, check out these 5 apps..

Listen to Story

• Students have had to learn Physics, Chemistry and Biology without a live teacher’s support in a science laboratory this past year.
• As they relied on virtual classes, a lot of them gained back the confidence and understood the concepts. 
• Here is a list of five apps that you must try before your science exam.

The past year has been quite challenging for both students and the faculty, especially when we talk about studying a compounded subject like Science. Initially, there was a sense of panic among students as they had to learn Physics, Chemistry and Biology without a live teacher’s support in a science laboratory.

As they relied on virtual classes, a lot of them gained back the confidence and understood the concepts.

But as we approach the annual exam season, there is apprehension amongst many students on how they would clarify last-minute doubts before the exam. For such problems, there are several EdTech apps available to simplify science.

These apps come in many forms, addressing everything from math and science problems to enhancing attention and concentration. They offer accessibility and support for users seeking to bolster their problem-solving capabilities. In addition to educational applications, many of these apps are also tailored to address challenges in industries like business and mental health or even to provide career guidance.

Key Takeaways

Recognizing problem-solving apps.

Today’s technology landscape is filled with mobile apps that aim to address various challenges we face daily. Recognizing essential problem-solving apps for iOS and Android devices can significantly improve our efficiency and simplify our lives.

Android Apps

By incorporating these problem-solving apps into daily routines, users can enhance their mental abilities and find solutions to everyday challenges more effectively. Focusing on the right technology and investing time in useful software will undoubtedly improve one’s overall problem-solving skills.

Educational Benefits of Problem-Solving Apps

Boosting learning skills.

Problem-solving apps provide a variety of educational benefits to users, aiding in the development of crucial learning skills. These apps target different aspects of learning, such as memory, reading, writing, and listening, by presenting engaging challenges and activities. As users navigate through these tasks, they gain valuable insights and ideas that contribute to their overall understanding of an array of subjects.

Enhancing Critical Thinking

Problem-solving apps also play a vital role in enhancing critical thinking abilities. By offering various challenges and exercises, these apps motivate users to employ creative thinking, logical reasoning, and decision-making skills. Through continuous practice and application, the users develop a deeper understanding of concepts and improve their ability to analyze and evaluate scenarios.

Moreover, apps like Educurious , which connects students with real-world mentors and incorporates the Common Core aligned curriculum, help students build their critical thinking abilities and problem-based learning skills in line with the 21st-century technology demands.

Applications in Math and Science

Apps for math problems.

One such resource is Mathway , which caters to a wide range of mathematical topics. Mathway enables students to input math problems, offering step-by-step solutions and explanations to further their understanding. The app even has a graphing feature for visual learners.

Applications for Science Tasks

To assist students with their science homework, websites like Educators Technology can offer a selection of math problem solver apps. These apps not only tackle math problems but also provide additional reinforcement for understanding scientific concepts.

In conclusion, utilizing these various apps and online tools can significantly improve students’ problem-solving abilities in both math and science domains, paving the way for academic success.

Enhancing Attention and Concentration

In addition to Lumosity, other apps are well-regarded for their positive influence on attention and concentration. For instance, Calm Sage lists several brain training apps to help improve memory and cognitive function. These apps provide fun, challenging exercises that test users’ problem-solving skills and logical thinking abilities while also identifying areas of improvement. Engaging in these activities can foster perseverance, allowing individuals to tackle tasks with greater determination and success.

Support and Help within Apps

Problem-solving apps rely on user reviews and feedback to constantly improve their features and functionalities. Therefore, it’s essential to have an efficient way for users to communicate their experiences, suggestions, and issues. Developers should ensure that they actively monitor feedback and provide prompt responses to users who might need assistance with the app.

When it comes to navigating through an app, a well-designed interface and smooth user experience will keep users engaged and motivated to solve problems. This includes logical menu structures, consistent design elements, and clear labeling for different sections or features. Visual aids, such as color-coding or iconography, can further help users find their way around the app, streamlining the overall problem-solving process.

Problem-Solving Apps for Business

Applications for business challenges.

Mobile applications are becoming increasingly useful in improving customer services by providing quicker query resolution and 24/7 support. AI-based chatbots, often embedded in mobile apps, can help businesses respond to customer queries and questions more efficiently, resulting in better customer satisfaction.

Mental Health Support through Apps

Apps for anxiety.

Anxiety can manifest in different ways, but common symptoms include constant worrying, restlessness, and even physical symptoms like rapid heartbeat or shortness of breath. The following apps aim to provide support and techniques for managing anxiety:

Apps for Depression

Popular puzzle and brain games, memory-boosting puzzle games.

A variety of memory-boosting puzzle games are available for those who wish to sharpen their cognitive skills. These games are designed to challenge the brain and improve memory, logic, and problem-solving abilities. Some popular memory-boosting puzzle games include:

The New York Times Crossword

Career guidance through problem-solving apps.

Braingle is a unique app that pushes the limits of mental sharpness through the use of riddles and visual illusions. By presenting different types of puzzles, Braingle encourages users to strengthen their reasoning and analytical skills, which could be beneficial in various aspects of career growth.

By using these problem-solving apps and adopting a methodical approach to tackling career challenges, individuals can pave the way for continuous growth and achievement. It is vital to remember that enhancing one’s problem-solving skills is a journey, requiring dedication and persistent effort.

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What is the 3-body problem, and is it really unsolvable?

The three-body problem is a physics conundrum that has boggled scientists since Isaac Newton's day. But what is it, why is it so hard to solve and is the sci-fi series of the same name really possible?

A rocket launch. Our nearest stellar neighbor. A Netflix show. All of these things have something in common: They must contend with the "three-body problem." But exactly what is this thorny physics conundrum?

The three-body problem describes a system containing three bodies that exert gravitational forces on one another. While it may sound simple, it's a notoriously tricky problem and "the first real worry of Newton," Billy Quarles , a planetary dynamicist at Valdosta State University in Georgia, told Live Science.

In a system of only two bodies, like a planet and a star, calculating how they'll move around each other is fairly straightforward: Most of the time, those two objects will orbit roughly in a circle around their center of mass, and they'll come back to where they started each time. But add a third body, like another star, and things get a lot more complicated. The third body attracts the two orbiting each other, pulling them out of their predictable paths .

The motion of the three bodies depends on their starting state — their positions, velocities and masses. If even one of those variables changes, the resulting motion could be completely different. 

"I think of it as if you're walking on a mountain ridge," Shane Ross , an applied mathematician at Virginia Tech, told Live Science. "With one small change, you could either fall to the right or you could fall to the left. Those are two very close initial positions, and they could lead to very different states."  

There aren't enough constraints on the motions of the bodies to solve the three-body problem with equations, Ross said. 

Related: Cosmic 'superbubbles' might be throwing entire galaxies into chaos, theoretical study hints

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Get the world’s most fascinating discoveries delivered straight to your inbox.

But some solutions to the three-body problem have been found. For example, if the starting conditions are just right, three bodies of equal mass could chase one another in a figure-eight pattern. Such tidy solutions are the exception, however, when it comes to real systems in space.

Certain conditions can make the three-body problem easier to parse. Consider Tatooine , Luke Skywalker's fictional home world from "Star Wars" — a single planet orbiting two suns. Those two stars and the planet make up a three-body system. But if the planet is far enough away and orbiting both stars together, it's possible to simplify the problem. 

"When it's the Tatooine case, as long as you're far enough away from the central binary, then you think of this object as just being a really fat star," Quarles said. The planet doesn't exert much force on the stars because it's so much less massive, so the system becomes similar to the more easily solvable two-body problem. So far, scientists have found more than a dozen Tatooine-like exoplanets , Quarles told Live Science.

But often, the orbits of the three bodies never truly stabilize, and the three-body problem gets "solved" with a bang. The gravitational forces could cause two of the three bodies to collide, or they could fling one of the bodies out of the system forever — a possible source of "rogue planets" that don't orbit any star , Quarles said. In fact, three-body chaos may be so common in space that scientists estimate there may be 20 times as many rogue planets as there are stars in our galaxy.

When all else fails, scientists can use computers to approximate the motions of bodies in an individual three-body system. That makes it possible to predict the motion of a rocket launched into orbit around Earth, or to predict the fate of a planet in a system with multiple stars.

— 'Mathematically perfect' star system being investigated for potential alien technology

— How common are Tatooine worlds?

— Mathematicians find 12,000 new solutions to 'unsolvable' 3-body problem

With all this tumult, you might wonder if anything could survive on a planet like the one featured in Netflix's "3 Body Problem," which — spoiler alert — is trapped in a chaotic orbit around three stars in the Alpha Centauri system , our solar system 's nearest neighbor. 

"I don't think in that type of situation, that's a stable environment for life to evolve," Ross said. That's one aspect of the show that remains firmly in the realm of science fiction.

Skyler Ware is a freelance science journalist covering chemistry, biology, paleontology and Earth science. She was a 2023 AAAS Mass Media Science and Engineering Fellow at Science News. Her work has also appeared in Science News Explores, ZME Science and Chembites, among others. Skyler has a Ph.D. in chemistry from Caltech.

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Game Central

Microsoft Math Solver 4+

Instant math explanations, microsoft corporation.

• 4.9 • 41.6K Ratings

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Description.

Math that’s easy, fun, and accessible! Math Solver is an app from Microsoft where every feature is 100% free (like step-by-step instructions) for learners of all ages and abilities. That means whether you’re practicing simple arithmetic or learning advanced algebra and calculus, we’ll guide you to the answer instantly. All for free and now with a games section! HOW IT WORKS: Use your camera to scan the problem, write it out on our whiteboard, or enter it in our calculator. Whatever you choose, Math Solver will help you solve it instantly with a detailed step-by-step explanation, interactive graphs, or similar problems from the web. And when you’re ready for more practice, you can take a daily quiz, play a game, or explore video lectures and articles. WHAT YOU CAN DO: ● Scan printed or handwritten math ● Write a math equation on screen like you would on paper ● Type and edit using advanced math calculator ● Get interactive step-by-step explanations ● Access and leverage our graphing calculator ● Import images with math equations from your mobile device ● Learn more with similar problems and video lectures ● Scan and plot x-y data tables for linear/non-linear functions ● Learn in your language – we support Chinese, French, German, Hindi, Italian, Japanese, Portuguese, Russian, Spanish, and more WHAT WE CAN SOLVE: ● Elementary: arithmetic, real, complex numbers, LCM, GCD, factors, and percentages ● Pre-Algebra: radicals and exponents, fractions, matrices, determinants ● Algebra: quadratic equations, system of equations, inequalities, rational expressions, linear, quadratic and exponential graphs ● Basic Calculus: summations, limits, derivatives, integrals ● Statistics: mean, median, mode, standard deviation, permutations, combinations Learn more about Microsoft Math Solver on our website: https://math.microsoft.com/ Contact us at [email protected] or @microsoftmath on social. Looking forward to your feedback!

Version 1.4.4

- Bug fixes and performance improvements

Ratings and Reviews

41.6K Ratings

i like this app because it shows the work and it absolutely free! you can type in the problem, show them problem and draw the problem! This app honestly amazing! It shows the work just in case you still don’t understand. This app made my life turn into depression to happiness. When I didn’t have this app I didn’t understand anything my professor was saying because I get distracted a lot, I had no clue on what to do on my papers, I would always fail and my mother would always get angry for failing which made me feel like a failure in life i could handle her screaming at me. Music was my therapy back then. I always cry every night because of my grades and everything that was happening to me. My best friend moved and didn’t say goodbye to me only to his other friends. My mom would always punish me by taking away my phone therefore i had no music and made my life worst, i would always stay up until 5-7 because i never wanted to wake up in the morning. I loved it when i was alone home. My dad was my favorite, he left me but i still go with him. I love him so much. Once i downloaded this app this cured all of my bad grades and my mom started to become more proud of me. So i hope you see this and download this!

So far so good

I’ve only been using this app for a short time, and so far it’s been great. You can write problems with your finger ( or stylist), type them in or take a picture of a problem (haven’t tested the picture on yet) but the first two work great. I had used Math Papa for many years and that was my go to…until they decided that if you want to see the step by step you have to pay a whopping $10 a month. This app shows how they got the answer for free. It amazes me that everyone acts so concerned about the U.S. being behind in math with the rest of the world, but people don’t want to give you good training for free. Kudos to Microsoft (for once) for free learning…that is, not charging to see the step by step solutions. May change this if anything changes after having used it longer.

Sure, I'd be happy to write a review for the Microsoft Math Solver app! The Microsoft Math Solver app is a fantastic tool for students and anyone who needs help solving math problems. The app is intuitive and easy to use, with a user-friendly interface that makes finding solutions to complex math problems a breeze. One of the standout features of the app is its ability to recognize handwritten math problems. This means that users can simply write out a problem on their device's screen using their finger or a stylus and the app will automatically recognize the problem and provide a step-by-step solution. In addition to handwritten recognition, the app also allows users to take a photo of a printed math problem and receive a solution. This feature is particularly useful for students who may not have access to a computer or who prefer to work on paper. Overall, the Microsoft Math Solver app is an excellent tool for anyone who needs help with math. Whether you're a student struggling with a difficult homework assignment or an adult trying to solve a complex math problem, this app is sure to be a helpful resource. Highly recommended!

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10 Hard Math Problems That Continue to Stump Even the Brightest Minds

Maybe you’ll have better luck.

For now, you can take a crack at the hardest math problems known to man, woman, and machine. For more puzzles and brainteasers, check out Puzzmo . ✅ More from Popular Mechanics :

• To Create His Geometric Artwork, M.C. Escher Had to Learn Math the Hard Way
• Fourier Transforms: The Math That Made Color TV Possible
• The Game of Trees is a Mad Math Theory That Is Impossible to Prove

The Collatz Conjecture

In September 2019, news broke regarding progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. And while the story of Tao’s breakthrough is promising, the problem isn’t fully solved yet.

A refresher on the Collatz Conjecture : It’s all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. Take any natural number, apply f, then apply f again and again. You eventually land on 1, for every number we’ve ever checked. The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity).

✅ Down the Rabbit Hole: The Math That Helps the James Webb Space Telescope Sit Steady in Space

Tao’s recent work is a near-solution to the Collatz Conjecture in some subtle ways. But he most likely can’t adapt his methods to yield a complete solution to the problem, as Tao subsequently explained. So, we might be working on it for decades longer.

The Conjecture lives in the math discipline known as Dynamical Systems , or the study of situations that change over time in semi-predictable ways. It looks like a simple, innocuous question, but that’s what makes it special. Why is such a basic question so hard to answer? It serves as a benchmark for our understanding; once we solve it, then we can proceed onto much more complicated matters.

The study of dynamical systems could become more robust than anyone today could imagine. But we’ll need to solve the Collatz Conjecture for the subject to flourish.

Goldbach’s Conjecture

One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude. But we need proof for all natural numbers.

Goldbach’s Conjecture precipitated from letters in 1742 between German mathematician Christian Goldbach and legendary Swiss mathematician Leonhard Euler , considered one of the greatest in math history. As Euler put it, “I regard [it] as a completely certain theorem, although I cannot prove it.”

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Euler may have sensed what makes this problem counterintuitively hard to solve. When you look at larger numbers, they have more ways of being written as sums of primes, not less. Like how 3+5 is the only way to break 8 into two primes, but 42 can broken into 5+37, 11+31, 13+29, and 19+23. So it feels like Goldbach’s Conjecture is an understatement for very large numbers.

Still, a proof of the conjecture for all numbers eludes mathematicians to this day. It stands as one of the oldest open questions in all of math.

The Twin Prime Conjecture

Together with Goldbach’s, the Twin Prime Conjecture is the most famous in Number Theory—or the study of natural numbers and their properties, frequently involving prime numbers. Since you've known these numbers since grade school, stating the conjectures is easy.

When two primes have a difference of 2, they’re called twin primes. So 11 and 13 are twin primes, as are 599 and 601. Now, it's a Day 1 Number Theory fact that there are infinitely many prime numbers. So, are there infinitely many twin primes? The Twin Prime Conjecture says yes.

Let’s go a bit deeper. The first in a pair of twin primes is, with one exception, always 1 less than a multiple of 6. And so the second twin prime is always 1 more than a multiple of 6. You can understand why, if you’re ready to follow a bit of heady Number Theory.

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All primes after 2 are odd. Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue. If a number is 3 more than a multiple of 6, then it has a factor of 3. Having a factor of 3 means a number isn’t prime (with the sole exception of 3 itself). And that's why every third odd number can't be prime.

How’s your head after that paragraph? Now imagine the headaches of everyone who has tried to solve this problem in the last 170 years.

The good news is that we’ve made some promising progress in the last decade. Mathematicians have managed to tackle closer and closer versions of the Twin Prime Conjecture. This was their idea: Trouble proving there are infinitely many primes with a difference of 2? How about proving there are infinitely many primes with a difference of 70,000,000? That was cleverly proven in 2013 by Yitang Zhang at the University of New Hampshire.

For the last six years, mathematicians have been improving that number in Zhang’s proof, from millions down to hundreds. Taking it down all the way to 2 will be the solution to the Twin Prime Conjecture. The closest we’ve come —given some subtle technical assumptions—is 6. Time will tell if the last step from 6 to 2 is right around the corner, or if that last part will challenge mathematicians for decades longer.

The Riemann Hypothesis

Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems , with $1 million reward for its solution. It has implications deep into various branches of math, but it’s also simple enough that we can explain the basic idea right here.

There is a function, called the Riemann zeta function, written in the image above.

For each s, this function gives an infinite sum, which takes some basic calculus to approach for even the simplest values of s. For example, if s=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using the imaginary number 𝑖—finding 𝜁(s) gets tricky.

So tricky, in fact, that it’s become the ultimate math question. Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.” On the plane of complex numbers, this means the function has a certain behavior along a special vertical line. The hypothesis is that the behavior continues along that line infinitely.

✅ Stay Curious: How to Paint a Room Using Math

The Hypothesis and the zeta function come from German mathematician Bernhard Riemann, who described them in 1859. Riemann developed them while studying prime numbers and their distribution. Our understanding of prime numbers has flourished in the 160 years since, and Riemann would never have imagined the power of supercomputers. But lacking a solution to the Riemann Hypothesis is a major setback.

If the Riemann Hypothesis were solved tomorrow, it would unlock an avalanche of further progress. It would be huge news throughout the subjects of Number Theory and Analysis. Until then, the Riemann Hypothesis remains one of the largest dams to the river of math research.

The Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is another of the six unsolved Millennium Prize Problems, and it’s the only other one we can remotely describe in plain English. This Conjecture involves the math topic known as Elliptic Curves.

When we recently wrote about the toughest math problems that have been solved , we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir Andrew Wiles solved it using Elliptic Curves. So, you could call this a very powerful new branch of math.

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In a nutshell, an elliptic curve is a special kind of function. They take the unthreatening-looking form y²=x³+ax+b. It turns out functions like this have certain properties that cast insight into math topics like Algebra and Number Theory.

British mathematicians Bryan Birch and Peter Swinnerton-Dyer developed their conjecture in the 1960s. Its exact statement is very technical, and has evolved over the years. One of the main stewards of this evolution has been none other than Wiles. To see its current status and complexity, check out this famous update by Wells in 2006.

The Kissing Number Problem

A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to the idea of stacking many spheres in a given space, like fruit at the grocery store. Some questions in this study have full solutions, while some simple ones leave us stumped, like the Kissing Number Problem.

When a bunch of spheres are packed in some region, each sphere has a Kissing Number, which is the number of other spheres it’s touching; if you’re touching 6 neighboring spheres, then your kissing number is 6. Nothing tricky. A packed bunch of spheres will have an average kissing number, which helps mathematically describe the situation. But a basic question about the kissing number stands unanswered.

✅ Miracles Happen: Mathematicians Finally Make a Breakthrough on the Ramsey Number

First, a note on dimensions. Dimensions have a specific meaning in math: they’re independent coordinate axes. The x-axis and y-axis show the two dimensions of a coordinate plane. When a character in a sci-fi show says they’re going to a different dimension, that doesn’t make mathematical sense. You can’t go to the x-axis.

A 1-dimensional thing is a line, and 2-dimensional thing is a plane. For these low numbers, mathematicians have proven the maximum possible kissing number for spheres of that many dimensions. It’s 2 when you’re on a 1-D line—one sphere to your left and the other to your right. There’s proof of an exact number for 3 dimensions, although that took until the 1950s.

Beyond 3 dimensions, the Kissing Problem is mostly unsolved. Mathematicians have slowly whittled the possibilities to fairly narrow ranges for up to 24 dimensions, with a few exactly known, as you can see on this chart . For larger numbers, or a general form, the problem is wide open. There are several hurdles to a full solution, including computational limitations. So expect incremental progress on this problem for years to come.

The Unknotting Problem

The simplest version of the Unknotting Problem has been solved, so there’s already some success with this story. Solving the full version of the problem will be an even bigger triumph.

You probably haven’t heard of the math subject Knot Theory . It ’s taught in virtually no high schools, and few colleges. The idea is to try and apply formal math ideas, like proofs, to knots, like … well, what you tie your shoes with.

For example, you might know how to tie a “square knot” and a “granny knot.” They have the same steps except that one twist is reversed from the square knot to the granny knot. But can you prove that those knots are different? Well, knot theorists can.

✅ Up Next: The Amazing Math Inside the Rubik’s Cube

Knot theorists’ holy grail problem was an algorithm to identify if some tangled mess is truly knotted, or if it can be disentangled to nothing. The cool news is that this has been accomplished! Several computer algorithms for this have been written in the last 20 years, and some of them even animate the process .

But the Unknotting Problem remains computational. In technical terms, it’s known that the Unknotting Problem is in NP, while we don ’ t know if it’s in P. That roughly means that we know our algorithms are capable of unknotting knots of any complexity, but that as they get more complicated, it starts to take an impossibly long time. For now.

If someone comes up with an algorithm that can unknot any knot in what’s called polynomial time, that will put the Unknotting Problem fully to rest. On the flip side, someone could prove that isn’t possible, and that the Unknotting Problem’s computational intensity is unavoidably profound. Eventually, we’ll find out.

The Large Cardinal Project

If you’ve never heard of Large Cardinals , get ready to learn. In the late 19th century, a German mathematician named Georg Cantor figured out that infinity comes in different sizes. Some infinite sets truly have more elements than others in a deep mathematical way, and Cantor proved it.

There is the first infinite size, the smallest infinity , which gets denoted ℵ₀. That’s a Hebrew letter aleph; it reads as “aleph-zero.” It’s the size of the set of natural numbers, so that gets written |ℕ|=ℵ₀.

Next, some common sets are larger than size ℵ₀. The major example Cantor proved is that the set of real numbers is bigger, written |ℝ|>ℵ₀. But the reals aren’t that big; we’re just getting started on the infinite sizes.

✅ More Mind-Blowing Stuff: Mathematicians Discovered a New 13-Sided Shape That Can Do Remarkable Things

For the really big stuff, mathematicians keep discovering larger and larger sizes, or what we call Large Cardinals. It’s a process of pure math that goes like this: Someone says, “I thought of a definition for a cardinal, and I can prove this cardinal is bigger than all the known cardinals.” Then, if their proof is good, that’s the new largest known cardinal. Until someone else comes up with a larger one.

Throughout the 20th century, the frontier of known large cardinals was steadily pushed forward. There’s now even a beautiful wiki of known large cardinals , named in honor of Cantor. So, will this ever end? The answer is broadly yes, although it gets very complicated.

In some senses, the top of the large cardinal hierarchy is in sight. Some theorems have been proven, which impose a sort of ceiling on the possibilities for large cardinals. But many open questions remain, and new cardinals have been nailed down as recently as 2019. It’s very possible we will be discovering more for decades to come. Hopefully we’ll eventually have a comprehensive list of all large cardinals.

What’s the Deal with 𝜋+e?

Given everything we know about two of math’s most famous constants, 𝜋 and e , it’s a bit surprising how lost we are when they’re added together.

This mystery is all about algebraic real numbers . The definition: A real number is algebraic if it’s the root of some polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√6 are algebraic numbers.

✅ Try It Yourself: Can You Solve This Viral Brain Teaser From TikTok?

All rational numbers, and roots of rational numbers, are algebraic. So it might feel like “most” real numbers are algebraic. Turns out, it’s actually the opposite. The antonym to algebraic is transcendental, and it turns out almost all real numbers are transcendental—for certain mathematical meanings of “almost all.” So who’s algebraic , and who’s transcendental?

The real number 𝜋 goes back to ancient math, while the number e has been around since the 17th century. You’ve probably heard of both, and you’d think we know the answer to every basic question to be asked about them, right?

Well, we do know that both 𝜋 and e are transcendental. But somehow it’s unknown whether 𝜋+e is algebraic or transcendental. Similarly, we don’t know about 𝜋e, 𝜋/e, and other simple combinations of them. So there are incredibly basic questions about numbers we’ve known for millennia that still remain mysterious.

Is 𝛾 Rational?

Here’s another problem that’s very easy to write, but hard to solve. All you need to recall is the definition of rational numbers.

Rational numbers can be written in the form p/q, where p and q are integers. So, 42 and -11/3 are rational, while 𝜋 and √2 are not. It’s a very basic property, so you’d think we can easily tell when a number is rational or not, right?

Meet the Euler-Mascheroni constant 𝛾, which is a lowercase Greek gamma. It’s a real number, approximately 0.5772, with a closed form that’s not terribly ugly; it looks like the image above.

✅ One More Thing: Teens Have Proven the Pythagorean Theorem With Trigonometry. That Should Be Impossible

The sleek way of putting words to those symbols is “gamma is the limit of the difference of the harmonic series and the natural log.” So, it’s a combination of two very well-understood mathematical objects. It has other neat closed forms, and appears in hundreds of formulas.

But somehow, we don’t even know if 𝛾 is rational. We’ve calculated it to half a trillion digits, yet nobody can prove if it’s rational or not. The popular prediction is that 𝛾 is irrational. Along with our previous example 𝜋+e, we have another question of a simple property for a well-known number, and we can’t even answer it.

Dave Linkletter is a Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas. His research is in Large Cardinal Set Theory. He also teaches undergrad classes, and enjoys breaking down popular math topics for wide audiences.

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