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

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?

The reader will probably see no other course but rather laborious trial,

and yet the answer may be found directly by very simple arithmetic and without any experimental extraction of a square root.

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

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $101$. -- Finding a Square
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $131$. Finding a Square