Henry Ernest Dudeney/Puzzles and Curious Problems/245 - Magic Fifteen Puzzle/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $245$

Magic Fifteen Puzzle

This is Loyd's famous $14$-$15$ puzzle,

in which you were asked to get the $14$ and $15$ in their proper order by sliding them about in the box.

It was, of course, impossible of solution.

I now propose to slide them about until they shall form a perfect magic square

in which the four columns, four rows and two diagonals all add up to $30$.

It will be found convenient to use numbered counters in place of the blocks.

What are your fewest possible moves?

Solution

Move the counters in the following order:

$12$, $8$, $4$, $3$, $2$, $6$, $10$, $9$, $13$, $15$, $14$, $12$, $8$, $4$, $7$, $10$, $9$, $14$, $12$, $8$, $4$, $7$, $10$, $9$, $6$, $2$, $3$, $10$, $9$, $6$, $5$, $1$, $2$, $3$, $6$, $5$, $3$, $2$, $1$, $13$, $14$, $3$, $2$, $1$, $13$, $14$, $3$, $12$, $15$, $3$

of $50$ moves in all.

This results in the square:

Variant Solution

Martin Gardner points out that if the $14$ and $15$ are in the correct serial order at the start, then a magic square can be achieved in $37$ moves:

$15$, $14$, $10$, $6$, $7$, $3$, $2$, $7$, $6$, $11$, $10$, $14$, $3$, $2$, $11$, $10$, $9$, $5$, $1$, $6$, $10$, $9$, $5$, $1$, $6$, $10$, $9$, $5$, $2$, $12$, $3$

This results in the square:

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $245$. -- Magic Fifteen Puzzle
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $371$. Magic Fifteen Puzzle