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