130. Finding the Square. Six numbers are given: 4784887, 2494651, 8595087, 1385287, 9042451, 9406087. It is known that the sum of three of them is a perfect square. Which are these numbers?

The reader will probably not see any other way but the tedious method of trial and error, and yet there is a direct solution to the problem using only simple arithmetic considerations and not requiring the extraction of square roots.
130. The sums of the digits of the six given numbers are respectively

\[
\begin{array}{rrrrrr}
46 & 31 & 42 & 34 & 25 & 34 \\
1 & 4 & 6 & 7 & 7 & 7
\end{array}
\]

By repeatedly summing the digits of these sums, we eventually obtain single-digit numbers, which are listed in the second row.

These single-digit numbers are called the digital roots of the original numbers. The digital roots can be grouped into sets of three numbers in eight different ways:

\[
\begin{array}{rrrrrrrr}
146 & 147 & 167 & 177 & 467 & 477 & 677 & 777 \\
2 & 3 & 5 & 6 & 8 & 9 & 2 & 3
\end{array}
\]

(At the bottom, the digital roots are listed.) As shown in my book "Mathematical Recreations," the digital root of a square must be \(1, 4, 7,\) or 9. Therefore, the sought numbers must have digital roots \(4, 7, 7\). The two sevens can be chosen in three ways. However, if the fifth number were among the sought ones, their sum would end in 189 or 389, which is impossible for a square, as an even number or 0 must precede 89. Therefore, the answer is

\[
2494651 + 1385287 + 9406087 = 13286025
\]

On the right side, we have a number that is the square of 3645.

To emphasize the value of this new method, I will quote Professor Rouse Ball:

"This application is entirely due to Mr. Dudeney. The properties of digits are little known to mathematicians, and we hope that his example will help to draw attention to this method... In solving certain classes of arithmetical problems, the method of digital roots is extremely useful."