Henry Ernest Dudeney/Puzzles and Curious Problems/185 - The Dissected Chessboard

Puzzles and Curious Problems by Henry Ernest Dudeney: $185$

The Dissected Chessboard

Here is an ancient and familiar fallacy.

If you cut a chessboard into four pieces in the manner indicated by the black lines in Figure $\text A$,

and then reassemble the pieces as in Figure $\text B$,

you appear to gain a square by the operation,

since this second figure would seem to contain $13 \times 5 = 65$ squares.

I have explained this fallacy over and over again, and the reader probably understands all about it.

The present puzzle is to place the same four pieces together in another way

so that it may appear to the novice that instead of gaining a square we have lost one,

the new figure apparently containing only $63$ cells.

It is best to use the diagram that is not chequered in black and white cells.

Also see

• Sam Loyd's Missing Square, upon which this entry was based

• Puzzles and Curious Problems: $189$ - Problem of the Extra Cell, where he raises the initial problem once more.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Geometrical Problems: Dissection Puzzles: $185$. -- The Dissected Chessboard
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Geometrical Problems: Dissection Puzzles: $353$. Problem of the Missing Cell