From Mathologer.

There is a lot more to the pretty equation 10² + 11² + 12² = 13² + 14² than meets the eye. Let me show you.

00:00 Intro
00:07 Animated visual proofs
03:35 Mathologer materializes
06:31 Three puzzles
07:45 Thanks!

Notes:
The beautiful visual proof for the squares pattern is based on a note by Michael Boardman in Mathematics Magazine: https://tinyurl.com/2d4y7wtf

As far as I can tell, I am the first one to notice that this beautiful argument also works for those consecutive integer sums (but I am probably wrong 🙂

I first read about the two patterns that this video is about in the 1966 book Excursions in Number Theory by Ogilvy and Anderson (pages 91 and 92).

The article "Consecutive integers having equal sums of squares" J.S. Vidger, Mathematics Magazine, Vol. 38, No. 1 (Jan., 1965), pp. 35-42. is dedicated to finding generalisations of the sort of equations that the squares pattern is all about. Here is a particularly, nice example derived at the very end of this article: 4² + … + 38² = 39² + … + 48². This article is on JSTOR https://www.jstor.org/stable/2688015.

I first encountered the Russian painting that puzzle 2 is about in an article by Ethan Siegel about 10² + 11² + 12² = 13² + 14² and Co. https://tinyurl.com/y7p5k4kw Nice find 🙂

365 is the smallest integer that can be expressed as a sum of consecutive square in more than one way 365 = 10² + 11² + 12² = 13² + 14² (and of course 365 also happens to be the number of days in a year 🙂 Viewer Exception2001: Knowing the result, it’s fun to think about making an efficient one-page calendar where the front is a 13×13 square and the back is a 14×14 square, with each square containing a date 😀

Viewer k k notes that consecutive squares also take care of leap years 🙂 8² + 9² + 10² + 11² = 366

Christofer Hallberg did some computer experiments and found the following beautiful equation: 4³+…+28³=30³+31³+32³+33³+34³

There are some nice families of equations involving sums of alternating sums of consecutive squares. Check out Roger Nelsen’s one glance proof https://tinyurl.com/2xauf83u
2² – 3² + 4² = -5² + 6²
4² – 5² + 6² – 7² + 8² = -9² + 10² -11² + 12²
…
Fun fact: the top part of the logo is the top part of the last image I show in the previous video https://youtu.be/94mV7Fmbx88?t=2547

Several viewers (Exception1, Nana Macapagal, B Smith, Shay) noticed that the projected cubes pattern differences are of the form n²(n+1)²/2 = 2(1 + 2 + 3 + … + n)².
5³ + 6³ = 7³ – 2
16³ + 17³ + 18³ = 19³ + 20³ – 18
33³ + 34³ + 35³ + 36³ = 37³ + 38³ + 39³ – 72
56³ + 57³ + 58³ + 59³ + 60³ = 61³ + 62³ + 63³ + 64³ – 200
85³ + 86³ + 87³ + 88³ + 89³ + 90³ = 91³ + 92³ + 93³ + 94³ + 95³ – 450
And that actually means that the nice visual proofs in the video do extend to these modified cubes pattern because the six slices of the cube that I show in the video actually do form the shell of a smaller cube LESS two diametrically opposed corners.

For the the 4th powers differences the formula is 4³(1 + 2 + 3 + … + n)³ = 8n³(n+1)³
7⁴ + 8⁴ = 9⁴ – 64
22⁴ + 23⁴ + 24⁴ = 25⁴ + 26⁴ – 1728
45⁴ + 46⁴ + 47⁴ + 48⁴ = 49⁴ + 50⁴ + 51⁴ – 13824
76⁴ + 77⁴ + 78⁴ + 79⁴ + 80⁴ = 81⁴ + 82⁴ + 83⁴ + 84⁴ – 64000
There is another nice piece of 4d hypercube geometry that goes with this observation.

The new emerging pattern then breaks again with 5th powers. Here the sequence of differences starts like this: 2002, 162066, 2592552, 20002600, 101258850, …. But who knows , … 🙂

The triangular numbers tn=1+2+3+…+n that feature prominently in all this arrange themselves into a nice pattern like this
t1+t2+t3=t4
t5+t6+t7+t8=t9+t10
t11+t12+…+t15=t16+t17+t18
etc.

Solving x² + (x+1)² = (x+2)² has two integer solutions. The first is 3 corresponding to 3² + 4² = 5². The second is -1 corresponding to (-1)² + 0² = 1². You also get a second solution for every other equation in the square pattern.
(-1)² + 0² = 1²
(-2)² + (-1)² + 0² = 1² + 2²
etc.

Donald Sayers and qyrghyz point out that there is a nice discussion of minimal dissections of 6³ into eight pieces that can be reassembled into a 3³ a 4³ and a 5³ in Martin Gardner’s book Knotted doughnuts and other mathematical entertainments, pages 198-200. A picture of a dissection like this is shown on this wiki page on Euler’s conjecture https://tinyurl.com/27pkbj2c Another dissection here https://tinyurl.com/y6c6tbj4

If you are interested in more Mathologer animations of the type shown at the beginning of this video check out Mathologer 2 and the final sections of many/most regular Mathologer videos.

T-shirt: google "super pi t-shirt"
Music: Here to fight by Roman P. and Earth, the Pale Blue Dot by Ardie Son

Enjoy!

Burkard