one plus one two https://oneplusonetwo.fun/ Small notes on arithmetic, puzzles and mental math. en Sun, 21 Jun 2026 10:30:00 +0000 Handshakes and triangular numbers https://oneplusonetwo.fun/posts/handshakes-and-triangles/ https://oneplusonetwo.fun/posts/handshakes-and-triangles/ Sun, 21 Jun 2026 10:30:00 +0000 If everyone at a party of <code>n</code> people shakes hands with everyone else exactly once, how many handshakes are there? Estimating square roots in your head https://oneplusonetwo.fun/posts/square-roots-by-estimate/ https://oneplusonetwo.fun/posts/square-roots-by-estimate/ Sun, 08 Mar 2026 10:30:00 +0000 To find √n, start from the nearest perfect square <code>a²</code>, and add the difference divided by <code>2a</code>: Monty Hall, counted by hand https://oneplusonetwo.fun/posts/monty-hall-by-hand/ https://oneplusonetwo.fun/posts/monty-hall-by-hand/ Sat, 17 Jan 2026 10:30:00 +0000 Three doors, one car. You pick door 1. The host, who knows where the car is, opens another door with a goat behind it and offers you a switch. Should you switch? The birthday problem without formulas https://oneplusonetwo.fun/posts/birthday-problem/ https://oneplusonetwo.fun/posts/birthday-problem/ Sat, 22 Nov 2025 10:30:00 +0000 In a room of 23 people, the chance that two share a birthday is just over one half. Most people guess something closer to 23/365, so it is worth seeing where the intuition goes wrong. A divisibility test for 7 that is actually usable https://oneplusonetwo.fun/posts/divisibility-by-7/ https://oneplusonetwo.fun/posts/divisibility-by-7/ Sat, 06 Sep 2025 10:30:00 +0000 Tests for 2, 3, 5, 9 and 11 are well known. The one for 7 is usually skipped because the textbook version is awkward. Here is a version that is easy to do in your head. Casting out nines https://oneplusonetwo.fun/posts/casting-out-nines/ https://oneplusonetwo.fun/posts/casting-out-nines/ Sat, 19 Jul 2025 10:30:00 +0000 A quick check for long sums and products that bookkeepers used for centuries before calculators. Squaring numbers that end in 5 https://oneplusonetwo.fun/posts/squares-ending-in-5/ https://oneplusonetwo.fun/posts/squares-ending-in-5/ Fri, 30 May 2025 10:30:00 +0000 Take the digits before the 5, multiply them by the next whole number, and write 25 after the result. Multiplying by 11 in your head https://oneplusonetwo.fun/posts/multiplying-by-11/ https://oneplusonetwo.fun/posts/multiplying-by-11/ Sat, 12 Apr 2025 10:30:00 +0000 This is the first trick most people learn, and it is still the one I use most often.