Henry Ernest Dudeney/Puzzles and Curious Problems/101 - Finding a Square/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $101$

Finding a Square
Here are six numbers:
$4 \, 784 \, 887$, $2 \, 494 \, 651$, $8 \, 595 \, 087$, $1 \, 385 \, 287$, $9 \, 042 \, 451$, $9 \, 406 \, 087$
It is known that three of these numbers added together will form a square.
Which are they?


Solution

The digital roots of each of the six numbers in order are:

$1, 4, 6, 7, 7, 7$

Combining these into triplets, and calculating the digital roots of each, we get:

$\begin {matrix}

146 & 147 & 167 & 177 & 467 & 477 & 677 & 777 \\ 

2  &  3  &  5  &  6  &  8  &  9  &  2  &  3 \end {matrix}$

From Digital Root of Square, every square number has a digital root in $\set {1, 4, 7, 9}$.

So the required numbers must have the digital roots $4$, $7$ and $7$ in order for their sum to be square.


Now, if the fifth number is included, then the total of the three numbers will end in $189$ or $389$.

This is impossible for a square number, as the $89$ would have to be preceded by an even digit.

Therefore the required numbers must be:

$2 \, 494 \, 651 + 1 \, 385 \, 287 + 9 \, 406 \, 087 = 13 \, 286 \, 025 = 3645^2$


Historical Note

W.W. Rouse Ball apparently commented on this puzzle as follows:

This application is original on Mr. Dudeney's part.
Digital properties are but little known to mathematicians, and we hope that his example may serve to direct attention to the method ... In a certain class of arithmetical problems it is of great assistance.


Sources

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