In math class (algebra 1), a classmate of mine realized this weird thing when asked the square or 21. In her head, knowing that $12^2$ = 144, she said 12 flipped is 21 so 144 flipped is 441, which is, in fact $21^2$. This doesn't work once you go past 14.

More examples (0)$1^2$ = (00)1 and $10^2$ = 100 $13^2$ = 169 and $31^2$ = 961

Why does this happen? Why doesn't it happen above 14? Are there other places this might work?

  • $\begingroup$ This just works for (some) small numbers because there is no carrying involved. It also works for longer numbers, like $112, 211$ or even $1112, 2111$ where no carrying is involved. $\endgroup$ – lulu May 21 at 22:45
  • $\begingroup$ Just a random comment: this was one of the first observations I made as a kid that got me into learning basic number theory! It's a great field. $\endgroup$ – Don Thousand May 21 at 22:49
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    $\begingroup$ I think you meant $441$ where you wrote $442$. $\endgroup$ – J. W. Tanner May 21 at 22:54
  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax. $\endgroup$ – dantopa May 21 at 23:00

You are squaring $10a+b$ vs. $10b+a$, and the squares are $100a^2+10\cdot2ab+b^2$ vs. $100b^2+10\cdot2ab+a^2$.

If $a^2$ and $b^2$ are a single digit ($0^2,1^2,2^2,3^2$) as well as $2ab$ (hence not $2\cdot3$), the swap works.


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