Henry Ernest Dudeney/Puzzles and Curious Problems/100 - Digital Squares/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $100$

Digital Squares
Find a number which, together with its square, shall contain all the $9$ digits once, and once only, the $0$ disallowed.
Thus, if the square of $378$ happened to be $152 \, 694$, it would be a perfect solution.
But unfortunately the actual square is $142 \, 884$, which gives us repeated $4$s and $8$s, and omits the $6$, $5$, and $9$.


Solution

The following solutions are the only ones:

\(\ds 567^2\) \(=\) \(\ds 321 \, 489\)
\(\ds 854^2\) \(=\) \(\ds 729 \, 316\)


Proof

See Penholodigital Square Equation for a full analysis.


This theorem requires a proof.
In particular: Revisit that page so as to ensure that Dudeney's own analysis is implemented: We need only examine cases where the digits in the root number sum to $9$, $18$, or $27$; or $8$, $17$, or $26$, and it can never be a lower sum than $317$ to form the necessary six figures.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
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$\blacksquare$


Sources

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