Little Things Matter


This is a very famous puzzle.  Here is motivation:

Infinite Chocolate (gif from

Now consider following variation.


Where does the hole in second triangle come from? (image credit: Rookie1Ja ,

The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is

\arctan(\frac{2}{3}) - \arctan( \frac{3}{8}) = \arctan (\frac{1}{46})

which is less than 1 degree 15′ . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the “hidden” lozenge is equal to that of a small square of the chessboard.


It looks like a triangle, because a thick line was used. Hypotenuse of the composite triangle is actually not a straight line – it is made of two lines. Forth cusps are where the arrows point (c9, l6).

Also there is an interesting video illustrating this in real life:


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