Two Geometry Brainteasers!

When writing fun stuff for my company’s global communications site, sometimes I like to throw in a brainteaser every now and again—especially given that a lot of PTC’s employees and customers are engineers. So here are two geometry puzzles to get those old mental juices flowing!

64 = 65 Geometry Paradox

Where does the hole in second triangle come from (the partitions are the same)?

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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 2/3 – arctan 3/8 = arctan 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.

Write Numbers

Write the numbers from 1 to 8 into the squares, so that the squares with consecutive numbers do not touch (neither edges nor corners).

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These and other amazing geometry puzzles can be found here on