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?
On
Any number only consisting of digits 0, 1, 2 & 3 and the sum of its first n digits being less than or equal to 2 + n will posses this property.
For example 23021, this number will not posses this property because the sum of the first 2 digits is 5 which is greater then 2 + 2 here n = 2
And the number 22021 will posses this property as it follows all the condition
For numbers ending with 0 like 2020
2020 ==> 0202
2020² = 4080400
The square of 0202 must be considered as 0040804
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.