Henry Ernest Dudeney/Modern Puzzles/173 - Difference Squares/Solution

Modern Puzzles by Henry Ernest Dudeney: $173$

Difference Squares

Can you rearrange the nine digits in the square so that in all the eight directions the difference between one of the digits and the sum of the remaining two shall always be the same?

$\qquad \begin{array}{|c|c|c|} \hline 4 & 3 & 2 \\ \hline 7 & 1 & 9 \\ \hline 6 & 5 & 8 \\ \hline \end{array}$

In the example shown it will be found that all the rows and columns give the difference $3$:

(thus $4 + 2 - 3$, and $1 + 9 - 7$, and $6 + 5 - 8$, etc.),

but the two diagonals are wrong, because $8 - \paren {4 + 1}$ and $6 - \paren {1 + 2}$ is not allowed:

the sum of the two must not be taken from the single digit, but the single digit from the sum.

How many solutions are there?

Solution

These may be the only ones:

$\qquad \begin{array}{|c|c|c|} \hline 2 & 1 & 4 \\ \hline 3 & 5 & 7 \\ \hline 6 & 9 & 8 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 8 & 1 & 4 \\ \hline 3 & 5 & 7 \\ \hline 6 & 9 & 2 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 2 & 1 & 6 \\ \hline 3 & 5 & 7 \\ \hline 4 & 9 & 8 \\ \hline \end{array}$

The difference throughout is $5$.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $173$. -- Difference Squares
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $386$. Difference Squares