Sam Loyd's Missing Square

Paradox

Consider a square of side length $13$.

Let it be divided into:

a $13 \times 5$ rectangle, divided into two right triangles by its diagonal
a $13 \times 8$ rectangle, divided into trapezia with side lengths $8$ and $5$:


Let the pieces be rearranged to form two long right triangles arranged to form a $21 \times 8$ rectangle.

The area of the square is $13 \times 13 = 169$.

The area of the rectangle is $21 \times 8 = 168$.

Where did the missing $1 \times 1$ square go?


Variant

You have a square which is made from $4$ large triangles, $4$ small triangles, $4$ irregular octagons and $4$ small squares.

You jumble them up and reassemble the pieces once again into that same large square, but this time there is a hole in the middle.

Where did that hole come from?


Resolution

This is a falsidical paradox.


When you place the $13 \times 5$ right triangle against the trapezium, supposedly to make a $21 \times 8$ right triangle, the hypotenuse of that figure is not actually straight.

When the $21 \times 8$ rectangle is drawn accurately, you will see an overlap:

It is noted that $5$, $8$ and $13$ are consecutive Fibonacci numbers.

From Cassini's Identity:

$F_n^2 = F_{n - 1} F_{n + 1} \pm 1$

Hence the resolution.

$\blacksquare$


Source of Name

This entry was named for Sam Loyd.


Historical Note

While puzzle has Sam Loyd's name attached to it, it appears to have been first published in $1774$ in Rational Recreations by William Hooper, about whom little seems to be known.


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
  • 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Vanishing Square Paradox: $143$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$
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