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For any positive integer n, there exists a circle that passes through exactly n lattice points in the Euclidean plane. This was first proven by Polish mathematician Andrzej Schinzel in 1958. It’s known as Schinzel’s theorem.

A puzzle by S. Dvoryaninov from the July-August 1994 issue of Quantum: A very large military band marched in square formation on a parade ground, then regrouped into a rectangle so that the number of rows increased by 5. How many musicians were in the band?

The 2024 GCHQ Christmas Challenge is now live. Devised by Government Communications Headquarters, the British intelligence agency, this year’s puzzles encourage children aged 11-18 to think laterally and work as a team, testing skills including codebreaking, mathematics, and lateral thinking. “This year for the first time there are three additional elements hidden within the card for those who want to take on an extra...

Henry George Liddell and Robert Scott’s Intermediate Greek-English Lexicon of 1851 contains a sobering entry: ραφανιδοω: to thrust a radish up the fundament, a punishment of adulterers in Athens In recalling this to friends at Christmas in 1972, historian John Julius Norwich wrote, “I’m sure it must once have been familiar to every schoolboy, and now that the classics are less popular than they...

Freezing in the Canadian arctic in 1821, John Franklin noted some telling effects of fatigue in his companions: I observed, that in proportion as our strength decayed, our minds exhibited symptoms of weakness, evinced by a kind of unreasonable pettishness with each other. Each of us thought the other weaker in intellect than himself, and more in need of advice and assistance. So trifling...