How can these squares mesh so well

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. Here is link to that note: 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 C. Stanley Ogilvy and John T. 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 s