Sam Loyd's Missing Square

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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 $5$:
MissingSquare1.png


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.

MissingSquareVariant.png

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:

MissingSquare2.png

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 Sam Loyd's Missing Square 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