The formula $HE=\sqrt{SHE}$ translates in German to $ER=\sqrt{SIE}$. Find the solution for both, where each letter represents a digit.

(These are two separate puzzles: digits represented by S and E don't have to be the same between languages.)


If $HE^2=SHE$ then

$HE$ (call it $x$ for convenience) has the property that $x=x^2$ mod 100. Hence $x$ is either 0 or 1 mod each of 4,25. We can't have either 0,0 or 1,1 because then we actually have $x^2=x$ which would mean $S=0$ (and also $H=0$). 0,1 yields $HE=76$ whose square is clearly too big. $1,0$ yields $HE=25$ which works ($SHE=625$).

If $ER^2=SIE$ then

clearly $0<E\leq3$ (else the square has too many digits) and $E$ is the last digit of a square so $E=1$. Then $R$ must be 1 (no!) or 9, leading to $ER=19$ and $SIE=361$.

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  • $\begingroup$ And here I thought they used the same rule... $\endgroup$ – AxiomaticSystem Jun 28 '19 at 21:37

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