Henry Ernest Dudeney/Puzzles and Curious Problems/177 - Square of Squares/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $177$

Square of Squares

Cutting only along the lines, what is the smallest number of square pieces into which the diagram can be dissected?

The largest number possible is, of course, $169$, where all the pieces will be of the same size -- one cell -- but we want the smallest number.

We might cut away the border on two sides, leaving one square $12 \times 12$, and cutting the remainder into $25$ little squares, making $25$ in all.

This is better than $169$, but considerably more than the fewest possible.

Solution

The fewest pieces must be $11$, and the solution (barring reflection of the section with the $8$ smaller squares) is unique.

Historical Note

This puzzle also appears in Dudeney's $1917$ collection Amusements in Mathematics, under the title $173$ - Mrs. Perkins's Quilt.

The problem had previously appeared in $1907$ in the first issue of Our Puzzle Magazine, edited by Sam Loyd.

This was later published in Loyd's pusthumous collection Cyclopedia of Puzzles.

If Loyd stole it from Dudeney, then Dudeney must have published it somewhere before $1907$.

Martin Gardner reports in $1968$ that the general problem of dividing a square lattice of any size along lattice lines into the smallest number of squares has been discussed but not solved in several places, including by him.

The corresponding problem on a triangular lattice, he reports, had not been studied at that time.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $177$. -- Square of Squares
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $343$. Square of Squares