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

by : $101$

 * Finding a Square

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