Patterns in the last two digits of Squares... Dimensions and Geometry

X 0 1 2 3 4 5 6 7 8 9
0 01 04 09 16 | 36 49 64 81
1 21 44 69 96 | 56 89 24 64
2 41 84 29 76 | 76 29 84 41
3 61 24 89 46 | 96 69 44 21
4 81 64 49 36 | 16 09 04 01

I am finding it a bit more informative to look at the numbers to see all the different ways the squares relate rather than just memorizing the equations.

The symmetry is nice. It would be interesting to play with how this is changed by adding dimensions

Base 10 divides into 2 dimensions nicely.

In 3 dimensions in base 10 is it the LCD of 30 that sets the length of the series?

How about hypercubes? Is the series length 20?

It is a shame we can’t think in more than 3 dimensions. I suspect that if we could use our spatial memory these regular shapes and their relationships might be a lot easier to understand.

Has any one attempted to understand algebra geometrically (rather than, the other way around) and is there any interesting insights to be had from the effort or is it just the ravings of a newbie?


It teaches a lot to look beyond the symmetry and ask yourself ‘why is this the way it is?’
Why do all numbers that end in 3 have a square that ends in 9, but also the numbers that end in 7?
Same with 4 and 6, 1 and 9 and 2 and 8?
Why is this?

I don’t see the same symmetry in 3 dimensions. How do you see this?


revisiting this random post and adding a better picture.


Do we miss the beginning of this thread?
It starts strange and I expect other forum members to have no idea what we are talking about…

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Last digit of a square is the same as it’s compliment 72 = (10-3)2 = 100 - 60 + 9;

Multiplying a by digit is the same as multiplying by 10 plus it’s compliment

Spend a little time looking at the picture and the patterns and thinking about complements and the first 100 squares start to be easier and easier to think about.

But then squares are just a special case of numbers like 21 and 29 with natural complements
or you can just start to see that any number with the same number of 10’s starts to look juicy
whether it is 21 and 26 546 because you can just sweep up the 27 20’s and add 6 at the end mostly reading from left to right which seems like the best way to find a winner.

the could just as easily be 121 and 126. 127 * 120’s + 6 15246

and really you can left to right 12 and 13 just as easily as 11.

I keep hoping I am going to learn to trust my complements as readily as I trust addition.
It is so incredibly handy but I always stumble and check the subtraction mentally instead of just accepting the complement which fits so much nicer with the difference of squares and differences in general.

Probably best to ignore my ravings and just notice that there is an easy peasy repeating pattern for memorizing the 2 digit squares and that this pattern spreads out into very friendly distributions if you like that kind of thing. Or you could just shake your head and laugh a the silly old guy.

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Too late. I generated a list of the primes up to 100 and noticed what you are saying about the pattern in the last two digits. I’d gotten as far as thinking that mod 5 arithmetic might explain it, as it does with the first and that there’s got to be something going in the 3rd digit too but I don’t see it.