these are the only perfect squares

[@pepebriguglio6125]

Why is a list of the first 24 squares (up to 23²) a sufficient list? Already at 24² we get a new set of last two digits, 76. So, when do we know that the list is complete?

[@gerryiles3925]

Yes, you actually need to keep going until you see them start to repeat again backwards after 25^2 (625): 26^2 (676 repeat of 24^2), 27^2 (729 repeat of 23^2) etc.

[@pepebriguglio6125]

@gerryiles3925 Ah, thanks, I didn't know. Nice fact!

[@ShaunakDesaiPiano]

The title should have been “these are the only absolutely perfect squares”.

[@disgruntledtoons]

I guess today's solution was not a good place to stop.

[@michaelruiz9451]

Excellent proof!

[@goodplacetostop2973]

Not today

[@howardthompson3543]

26^2 -> 76 Note that this does not effect the (gist of the) proof

[@Bodyknock]

<a href="#" class="seekto" data-time="209">3:29</a> Also 10 should have been boxed since 00 swapped is still 00. (It’s technically the same as 0 being boxed.)

[@Bodyknock]

You can maybe streamline the solution a bit by noting that all the digits must be included in the list of digits that are possible for squares mod 100, i.e. all the digits are either 0,1,4,6, or 9.

[@acelm8437]

Happy new square year! (2025 = 45^2)

[@txikitofandango]

24^2 = 26^2 = 76 (mod 100) but the point still stands because 67 is not on the list

[@maxhagenauer24]

Yeah 24^2 totally equals 26^2...

[@Hiltok]

@@maxhagenauer24 mod 100 it totally does. In fact, (25-n)² is congruent to (25+n)² mod 100. I don't know why Michael was sloppy with this (other than to generate comments that feed the algorithm). (25-n)² = 625 - 50n + n² (25+n)² = 625 + 50n + n² (25+n)² - (25-n)² = 100n The same symmetry is in play for squares of 50+/-n and 75+/-n, so the squares of 1 to 24 do indeed cover all possible cases for the last two digits.

[@maxhagenauer24]

@Hiltok I didn't know he was talking about mod 100 and congruence, he still said 24^2 = 26^2. But (25 - n)^2 is not congruent to (25 + n)^2 if n is not a natural number. I wish Michael is the video was less sloppy as well and more regular and "true" because it wasn't at some parts.

[@Hiltok]

@@maxhagenauer24 Convention is 'n' is used for variable name when it is a natural number, just as we use 'x' for variable name when it is a real number. Most people reading maths will know than 'n' means a natural number (or integer) without it being explicitly stated.

[@maxhagenauer24]

@Hiltok That's not always true though, I seen people use n as any real number a couple times but it's generically used as a natural number while x is anything like you said.

[@petersiracusa5281]

In case 1, i miss why checking 0a0 is exhaustive.

[@KeimoKissa]

Is n is congruent to 0 mod 100, it will end in 00. For the number to be absolutely square, every permutation, including 0a0, will have to be a square. If it's not, not every permutation is a square, so the number can't be absolutely square.

[@supratimsantra5413]

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[@bwoy12345]

I dont get it.

[@fernandobeiroa]

16*16=256 25*25=625

[@millwrightrick1]

How about 526? Or 652? They don't work.

[@Hiltok]

That is only two of the six permutations of 2,5,6. What about 526, 562, 265, 652?