Henry Ernest Dudeney/Puzzles and Curious Problems/189 - Problem of the Extra Cell/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $189$

Problem of the Extra Cell

In diagram $A$ the square representing a chessboard is cut into $4$ pieces along the dark lines,

and these four pieces are seen re-assembled in Diagram $B$.

But in $A$ we have $64$ of these little squares, whereas in $B$ we have $65$.

Where does the additional cell come from?

Solution

Also see

• Sam Loyd's Missing Square

• Puzzles and Curious Problems: $185$ - The Dissected Chessboard, where the question is raised again from a different angle.

Historical Note

Martin Gardner discusses this and several other paradoxes of a similar type in his Mathematics, Magic and Mystery from $1956$, in chapters $7$ and $8$: Geometrical Vanishes, parts $\text I$ and $\text {II}$.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $189$. -- Problem of the Extra Cell
• 1956: Martin Gardner: Mathematics, Magic and Mystery: Chapter $8$ - Geometrical Vanishes -- Part $\text {II}$: Square Variation
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $352$. Problem of the Extra Cell