Solid Copper Coin

• Instructions for how to use this coin included
• Solid copper
• 1" (2.5 cm) coin weighs 16.3 grams
• Expert patina to highlight the engraving

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A post shared by physicsfun (@physicsfun)

The first documented description of this geometric brain teaser comes in the form of a magic trick by Elias et al. under the title Screwy Square published in 1946. A multitude of variations have since been developed and described, and most involve a square or octagon cutout with an arrow printed on each face, always incorporating the following main important feature: the two arrows are at a 90° right angle with respect to each other.

In his booklet of dinner table magic, Over the Coffee Cups (1949), Martin Gardner described a variation entitled Crazy Crackers, to whom he credits Val Evans. This version featured a marking pen and a square soda cracker, with which Gardner advises, once your audience is astonished “crumble it up before anyone has a chance to experiment with it”. Along a similar timeline, the English magician Willane published a version he called The Chinese Compass in his volume Willane's Wizardry (1947), which was the first to established the octagon presentation of the puzzle.

As described, The Magic Octagon involves two vectors (arrows) that share a fixed angle of 90°, and since each is imprinted on the two sides of a coin, only one arrow’s direction can be viewed at a time. The challenge is to determine the orientation of the second arrow when the first arrow’s orientation is changed, and this determination involves considering three separate rotations: two vertical flips, and one 45° rotation about a centered axis perpendicular to the face of the coin. It turns out that this is not something that comes easily to the brains of a majority of us, and while science confirms this deficit, the reason for it remains a current subject of research in cognitive psychology.

However, the fact that most are quite surprised by the outcome makes this “discrepant event” a valuable teaching tool in the geometry classroom-- virtually guaranteeing a raucous and robust classroom investigation as well as a memorable learning experience, and providing a confirmation that the angle between two vectors is indeed preserved under rotations.

Coin designed in collaboration with Raymond Hall (@physicsfun) and engraved by Shire Post Mint.

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