Category: Mathematics

  • 1+2+3+… = -1/12?

    John F. Lindner

    In quantum electrodynamics, the bare charge of an electron is infinite, but the renormalized dressed charge is finite. The bare electron shields itself by polarizing the virtual electron-positron pairs of the nearby quantum vacuum to reduce its coupling at large distances to in natural units, where the “I” decorating the equals sign denotes an informal relationship. Renormalization techniques help…

  • Math Grenade

    John F. Lindner

    I just bought a new calculator. New to me, that is, but older than me. Inspired by the 1600s Gottfried Leibniz stepped cylinder and the 1800s Thomas de Colmar arithmometer, the Curta mechanical calculator design was developed by Curt Herzstark while imprisoned in the Buchenwald concentration camp during World War II. Curta calculators were manufactured between…

  • The Impossible Problem

    John F. Lindner

    In 1969, Hans Freudenthal posed a puzzle that Martin Gardner would later call “The Impossible Problem”. Below is a 2000 version due to Erich Friedman. I have secretly chosen two nonzero digits and have separately told their sum to Sam and their product to Pam, both of whom are honest and logical. Pam says, “I…

  • Rubik’s Cube Puzzles

    John F. Lindner

    As a kid, I enjoyed solving the “15 puzzle”, a sliding puzzle consisting of a 4×4 grid of 15 squares. However, I was amazed by a kind of 3D analogue of the 15 puzzle: Ernö Rubik’s 1974 masterpiece, which is both a seemingly impossible mechanism (how does it not fall apart?) and a silent challenge (one…

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