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Leonhard Euler: 9 puzzles worth knowing

9 puzzles from Leonhard Euler, each explained in a minute and credited to the original work. Free on Savvy.

• The seven bridges of Königsberg

  The old city of Königsberg had seven bridges linking two islands and both riverbanks. Citizens tried for years to find a walk crossing every bridge exactly once. Can it be done?

  — Leonhard Euler, Solutio problematis ad geometriam situs pertinentis

• Euler's 36 officers who could never be arranged in a square

  Take 6 army regiments, each sending 6 officers of 6 different ranks. Can all 36 officers be arranged in a 6x6 square so that every row and column contains one officer of each rank …

  — Leonhard Euler, Recherches sur une nouvelle espèce de quarrés magiques

• A knight can visit every square on a chessboard exactly once

  Can a knight, moving only in its usual L-shape, tour an entire 8x8 chessboard, landing on every square exactly once? Indian and Arabic scholars had explored the pattern for centuri…

  — Leonhard Euler, Solution d'une question curieuse qui ne paroit soumise à aucune analyse

• Add up the reciprocal of every square number, forever. What do you get?

  Take 1 + 1/4 + 1/9 + 1/16 + 1/25 and so on, adding the reciprocal of every perfect square out to infinity. The sum obviously keeps growing more slowly and seems to approach some fi…

  — Leonhard Euler, De Summis Serierum Reciprocarum

• Euler's 1751 question to a friend quietly invented the Catalan numbers

  In 1751, Leonhard Euler wrote to his friend Christian Goldbach with a geometry question: cutting a convex polygon into triangles using only non-crossing diagonals, how many differe…

  — Leonhard Euler, Letter to Christian Goldbach, September 4, 1751

• Count a shape's corners, minus its edges, plus its faces. The answer is always the same.

  Take any convex polyhedron, a cube, a pyramid, a soccer-ball shape, and count its vertices (V), edges (E) and faces (F). Try computing V minus E plus F for a few different shapes b…

  — Leonhard Euler, Elementa Doctrinae Solidorum

• At any party, prove two people always know the exact same number of others

  At a party of any size, some guests shake hands with some others and not with the rest. Prove that no matter how the handshakes happen to fall, there must always be at least two pe…

  — Leonhard Euler, Handshaking lemma

• Find a number that exactly equals the sum of its own divisors

  Six is divisible by 1, 2, and 3 — and 1+2+3 happens to equal 6 exactly. The next such number is 28 (1+2+4+7+14=28). These are called perfect numbers. Is there a way to generate eve…

  — Euclid; Leonhard Euler (Euclid–Euler theorem), Perfect number

• Adding up the reciprocals of every square number gives a surprising answer

  Add 1/1² + 1/2² + 1/3² + 1/4² and so on, forever. The sum clearly settles on some finite number — but what number? Mathematicians across Europe failed to find it for nearly a centu…

  — Leonhard Euler, De Summis Serierum Reciprocarum

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