# Math

There’s an incredible method to calculate the square root of a perfect square in your head. MIT Entrace Exam Arithmetic 1876 WayBackMachine: https://web.archive.org/web/20190404065722/https://libraries.mit.edu/archives/exhibits/exam-entrance1876/arithmetic1876.html Square root method Burning Math http://burningmath.blogspot.com/2013/09/extracting-square-roots-mentally-at.html Square root method proof Math StackExchange https://math.stackexchange.com/questions/2069192/why-does-this-method-for-finding-the-square-root-of-a-number-work Square root method Grey Matters http://gmmentalgym.blogspot.com/2010/10/root-extractions.html#rtsqr Hindu-Arabic numeral system https://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numeral_system Laplace quote https://en.wikipedia.org/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system#Impact_on_arithmetic Laplace picture https://en.wikipedia.org/wiki/File:Laplace,_Pierre-Simon,_marquis_de.jpg Abacus picture https://en.wikipedia.org/wiki/Abacus#/media/File:Houghton_Typ_520.03.736_-_Margarita_philosophica.jpg

Thanks to Francisco from Portugal for the suggestion! They say Euler asked this tricky algebra problem to his students. Reference Euler’s Riddle Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address in video). I consider all ideas though can’t always reply! Like many YouTubers I use popular software to prepare my videos. You can search for

Thanks to Matej for suggesting this problem! This is from a math competition for 14 year olds in Croatia. While the problem is challenging, “you should be able to solve it” using elementary geometry and algebra. Can you figure it out? Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Playlist to watch all videos on MindYourDecisions: https://www.youtube.com/playlist?list=UUHnj59g7jezwTy5GeL8EA_g This is the only

In the past 5 months, all of Mark Rober’s videos have been #1 trending on YouTube. He put his success in perspective by analogy to a statistical experiment of flipping coins. Some people pointed out the calculation wasn’t quite correct, so I thought it would be a great topic for a video. Mark Rober tweet

In 1880, James Garfield contributed a new proof of geometry’s most famous right triangle theorem. Mathematical treasure: Garfield’s proof https://www.maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theorem Engraved portrait https://en.wikipedia.org/wiki/File:GARFIELD,_James_A-President_(BEP_engraved_portrait).jpg Gizmodo post https://gizmodo.com/james-garfield-was-the-only-u-s-president-to-prove-a-m-1037750658 Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas! If you purchase through these links,

This is a standard question to prepare for India’s famously challenging JEE Advanced exam. Here is one way to solve the question quickly. Thanks to Avinash for the suggestion! Special thanks this month to: Daniel Lewis, Robert Zarnke, Kyle, Mike Robertson. Thanks to all supporters on Patreon! http://www.patreon.com/mindyourdecisions Problem credit to David Altizio (section 3

Why five-sided figures pose a problem from Professor John Hunton – and a bit about the importance of Penrose Tiling. More links & stuff in full description below ↓↓↓ Professor Hunton works at the University of Leicester. NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Videos by Brady Haran Patreon: http://www.patreon.com/numberphile

A tetrahedron and a pyramid have edges of equal length. If they are glued together on a triangular face with the vertices aligned, how many faces will the new shape have? This question is trickier than it looks! It stumped the test-makers. Playable demonstration: Steve Phelps has made a Geogebra worksheet that you can try

Can you solve this problem that stumped many advanced math students? References https://io9.gizmodo.com/ready-this-simple-puzzle-once-stumped-96-of-americas-1698814691 https://www.theguardian.com/science/2017/jun/05/did-you-solve-it-are-you-in-the-smartest-10-per-cent https://www.mapleprimes.com/posts/206713-Puzzle-Or-A-Simple-Exercise http://puzzles.nigelcoldwell.co.uk/seventytwo.htm https://timss.bc.edu/timss1995i/TIMSSPDF/C_full.pdf Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address in video). I consider all ideas though can’t always reply! Like many YouTubers I use popular software to prepare my videos. You can search for animation software tutorials on YouTube

Thanks to Juanci for the suggestion! Also thanks to everyone who shared a solution on Twitter. A special thanks to LuisAndrRomero2 who provided the first detailed solution. Special thanks this month to: Michael Anvari, Kyle. Thanks to all supporters on Patreon! http://www.patreon.com/mindyourdecisions Who can solve this one? I'm working on a video for it and

The Big Bang Theory TV is now part of math history: there’s a theorem named after Sheldon. Sources Siouxland News article https://siouxlandnews.com/news/local/morningside-college-professors-math-appearing-in-the-big-bang-theory 73 is the best number – Big Bang Theory clip Proof of the Sheldon conjecture https://math.dartmouth.edu/~carlp/sheldon02132019.pdf 73 is the best number (other properties) Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Playlist to watch all videos on MindYourDecisions: https://www.youtube.com/playlist?list=UUHnj59g7jezwTy5GeL8EA_g

Colin Rizzio became famous after finding a mistake on the S.A.T. Sources https://www.deseret.com/1997/2/7/19293720/math-question-on-the-sat-didn-t-sit-well https://people.com/archive/flunking-the-tester-vol-47-no-7/ https://www.chicagotribune.com/news/ct-xpm-1997-02-07-9702070268-story.html https://www.dartmouth.edu/~chance/chance_news/recent_news/chance_news_6.03.html Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address in video). I consider all ideas though can’t always reply! Like many YouTubers I use popular software to prepare my videos. You can search for animation software tutorials on YouTube to

Thanks to Amit from India for the suggestion! A circle is tangent on the exterior of a side of a triangle and tangent to the lines through the other two sides. The tangent side length 7, and the other side lengths are 8 and 9. What is the radius of the circle? 0:00 Problem 1:22

Thanks to Dhrubajyoti for the suggestion! A similar problem was asked to class 10 students in India (Q33 from SMO junior 2016). I admit that this one stumped me. Can you figure it out? Solution presented: post by Thomas Hill on Quora https://www.quora.com/In-a-triangle-ABC-AB-AC-angle-BAC-100-circ-Let-D-be-a-point-such-that-B-is-an-interior-point-of-AD-and-AD-BC-How-can-you-find-angle-BDC Other solutions http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Rapley/EMT725/100deg/100deg.html http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Wise/essay1/essay1.htm https://gogeometry.blogspot.com/2011/11/problem-685-isosceles-triangle-100-40.html https://twitter.com/jldavilaa01/status/895386114390970368 Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by

People around the world have asked me this and similar problems. Learn how to solve using only geometry (and a little bit of algebra). There are many other ways to solve the problem too https://www.mathalino.com/reviewer/plane-geometry/02-area-common-arcs-quarter-circles Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address in video). I consider all ideas though can’t always reply! Like many

Large factorials and the use of Stirling’s Approximation. Featuring Professor Ken McLaughlin. More links & stuff in full description below ↓↓↓ Professor McLaughlin is based at Colorado State University: https://www.math.colostate.edu/~kenmcl/ We filmed this during his time at the Mathematical Sciences Research Institute. 69 Factorial: https://www.youtube.com/watch?v=kw6l_uTakRA Numberphile is supported by the Mathematical Sciences Research Institute (MSRI):

Thanks to Ryan from India for the suggestion! Two candles of equal height but different thickness are lit. The first burns off in 8 hours and the second in 10 hours. When will the first candle be half the height of the second? Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address at end of many

A 6’28” version of our “Pi with Pies” calculation. More links & stuff in full description below ↓↓↓ Videos features Matt Parker who tweets at https://twitter.com/standupmaths The Pi-inspired music in the video explained: http://periodicvideos.blogspot.co.uk/2013/03/pimusic.html Our full collection of Pi videos at http://bit.ly/W4oDN1 While the pies are usually delicious, this batch was not for consumption (even

There are 25 mechanical horses and a single racetrack. Each horse completes the track in a pre-programmed time, and the horses all have different finishing times, unknown to you. You can race 5 horses at a time. After a race is over, you get a printout with the order the horses finished, but not the

A compilation of Numberphile sound checks – Brady typically asks “what did you have for breakfast” so he can monitor audio levels. More links & stuff in full description below ↓↓↓ Previous sound checks with scientists here: http://youtu.be/4anqDfgQneA Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub

17 is the minimum number of clues required to give a unique sudoku solution – but how did mathematicians prove this? Featuring James Grime. More links & stuff in full description below ↓↓↓ Dr James Grime discusses a recent paper which cracked the problem. The paper being discussed by McGuire and others is at http://arxiv.org/abs/1201.0749

Thanks to Nikhil Patro for suggesting this problem! What is the sum of the corner angles in a regular 5-sided star? What if the star is not regular, what is the sum of corner angles then? There are some neat ways to solve this. I present my favorite methods. Thanks to all patrons! Special thanks

Our Pi Playlist (more videos): http://bit.ly/PiPlaylist How accurately can we calculate Pi using hundreds of REAL pies? More links & stuff in full description below ↓↓↓ Extended version of this video (director’s slice) at http://www.youtube.com/watch?v=x4kyFKyCMv0 This video features Matt Parker: https://twitter.com/standupmaths Matt believes this is the world’s most accurate pie-based Pi calculation. More Pi videos:

Thanks to Victor for the suggestion! A version of this problem appeared on a Kangaroo math competition for 12-13 year olds. Can you figure it out? Special thanks this month to: Michael Anvari, Daniel Lewis, Robert Zarnke, Kyle, Mike Robertson. Thanks to all supporters on Patreon! http://www.patreon.com/mindyourdecisions Original Math Kangaroo question on YouTube Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1

Matt Parker explores the work of William Shanks – and boots up the ShanksBot. More links & stuff in full description below ↓↓↓ Matt Parker’s 2022 Pi Day Video: https://youtu.be/dtiLxLrzjOQ Discussing William Shanks on Objectivity: https://youtu.be/7yTXMeiVBCc Prime Number playlist: https://bit.ly/PrimePlaylist Pi playlist: http://bit.ly/PiPlaylist Matt Parker website: https://standupmaths.com Numberphile is supported by the Mathematical Sciences Research

What is the radius of the circle, in simplest terms involving a, b, and c? I read that even the great mind Richard Feynman made a mistake. Watch the video to learn the common mistake and what the correct answer is. Source Feynman story http://web.media.mit.edu/~walter/MAS-A12/week11.html My blog post for this video https://wp.me/p6aMk-84E Playlist to watch

Cliff Stoll is passionate about Klein Bottles. More links & stuff in full description below ↓↓↓ Don’t miss the video about how he uses a robot to store 1,000 bottles UNDER his house… https://youtu.be/-k3mVnRlQLU More videos on Klein Bottles: http://bit.ly/KleinBottles ACME Klein Bottles: http://bit.ly/ACME_Klein Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook:

Another pass at the Monty Hall Problem – see the last video and a new “express explanation” at: http://bit.ly/MontyHallProb More links & stuff in full description below ↓↓↓ Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Numberphile is supported by the Mathematical Sciences Research Institute (MSRI):

Prime numbers, Ulam Spirals and other cool numbery stuff with Dr James Grime. More links & stuff in full description below ↓↓↓ James Clewett on spirals at: http://youtu.be/3K-12i0jclM And more to come soon… * subscribing to numberphile does not really change your physical appearance! And “golden line” in this context was made up by Brady!

When is n² + 45 equal to a perfect square, for n a positive integer? I received this MathCounts problem by email from Bill by email. Can you figure it out? Watch the video for a solution. My blog post for this video https://wp.me/p6aMk-5i0 If you like my videos, you can support me at Patreon:

Our Pi Playlist (more videos): http://bit.ly/PiPlaylist Steve Mould (tau) and Matt Parker (pi) go head-to-head. More links & stuff in full description below ↓↓↓ See our tau video: http://www.youtube.com/watch?v=83ofi_L6eAo And check our Pi videos: http://www.youtube.com/playlist?list=PL4870492ACBDC2E7C Matt Parker: http://www.standupmaths.com/ Steve Mould: http://www.stevemould.com/ NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Videos by

This video gives a method to convert decimal numbers to binary numbers quickly. This is a variation of the remainder system that is typically taught in courses. I like this method because it gives the answer in the correct order from left to right. Proof of why it works: http://math.stackexchange.com/questions/86207/converting-decimalbase-10-numbers-to-binary-by-repeatedly-dividing-by-2 I also give a proof

Catch a more in-depth interview with Ben Sparks on our Numberphile Podcast: https://youtu.be/-tGni9ObJWk Check out Brilliant (and get 20% off) by clicking https://brilliant.org/numberphile More links & stuff in full description below ↓↓↓ Golden seeds limited edition T-Shirt: https://teespring.com/NP-Seeds More Golden Ratio stuff: http://bit.ly/Golden_Ratio More Ben Sparks Numberphile videos: http://bit.ly/Sparks_Playlist Ben’s Twitter: https://twitter.com/SparksMaths Ben’s website: www.bensparks.co.uk

Many people asked me to solve this one! What is the value of x? Many more solutions Can anyone solve this? I have a numerical answer using trigonometry. I did find a geometry "proof" online but it was faulty. Many people are requesting this problem, so I would like to make a video. Help is

The harmonic series and the elusive Euler–Mascheroni constant. More links & stuff in full description below ↓↓↓ Featuring Dr Tony Padilla. Audible: http://www.audible.com/numberphile Extra footage: https://youtu.be/eRGN8ThZfhU Videos about -1/12: http://bit.ly/minus_twelfth Tony at the LHC: https://youtu.be/sVYUqMRolaA (via Sixty Symbols, our physics channel) Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets:

Tadashi Tokieda is back. This time talking about stability, instability and train wheels. More Tadashi videos: http://bit.ly/tadashi_vids Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile

A semicircle contains an inscribed semicircle dividing its diameter into two lengths a and b. Can you find the formula for the inscribed semicircle’s diameter in terms of the lengths a and b? What is the locus of the center of the inscribed semicircle? Thanks to Nick from Greece for the suggestion! Special thanks this

A ‘math metal’ song about the golden ratio (aka Phi). More links & stuff in full description below ↓↓↓ See more about the making of the song and maths behind it at: http://www.youtube.com/watch?v=mqK63v2Jzks This was a collaboration with the excellent Dave Brown – his channel is: http://www.youtube.com/user/wwwboyinabandcom (check it out!!!) Also featuring Professor Phil Moriarty

This is a fun little geometry puzzle! adapted from a puzzle by @Cshearer41 What’s the area of the toppled square? pic.twitter.com/77yHacED5Y — Catriona Agg (@Cshearer41) August 7, 2018 solution 1- area multiplication right squares: 3*4 = 12 and 3*9 = 27, that means *2 and *3 for the sides. 2- roots for sides: √3, 2√3

Solve this to get into Oxford! 2021 Oxford MAT Questions https://www.maths.ox.ac.uk/system/files/attachments/test21.pdf 2021 Oxford MAT Solutions https://www.maths.ox.ac.uk/system/files/attachments/websolutions21.pdf Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas! If you buy from the links below I may receive a commission for sales. (As an

Hannah Fry on parallels between the game “rock paper scissors” and lizards in nature. PART ONE: http://youtu.be/rudzYPHuewc More links & stuff in full description below ↓↓↓ EXTRA FOOTAGE: http://youtu.be/ygHwBxWyI6E Lizards by Pete McPartlan Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Numberphile is supported by the

These are the answers to the previous video with 3 matchstick puzzlers. If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you. My

Alex’s book: http://amzn.to/1l0yX46 The Curta is a pocket-sized, mechanical, digital calculator!!! It was invented by Curt Herzstark. More links & stuff in full description below ↓↓↓ Shown here by Alex Bellos, author of Alex’s Adventures in Numberland. More about our contributors, including Alex, at http://www.numberphile.com/team/index.html NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile

Thanks to Lucas for suggesting this problem! It comes from a Belgian Olympiad problem for 16 to 18 year old students. Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address in video). I consider all ideas though can’t always reply! Like many YouTubers I use popular software to prepare my videos. You can search for animation

How do you measure the length of a spiral on a cylinder? There is a neat trick! If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is

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