Category: games

  • Mamuth to Elephant (2)

    Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$-gon. If we move clockwise along the $12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$-vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the […]

  • From Mamuth to Elephant

    Here, MaMuTh stands for Mathematical Music Theory which analyses the pitch, timing, and structure of works of music. The Elephant is the nickname for the ‘bible’ of topos theory, Sketches of an Elephant: A Topos Theory Compendium, a two (three?) volume book, written by Peter Johnstone. How can we get as quickly as possible from…

  • Hexboards and Heytings

    A couple of days ago, Peter Rowlett posted on The Aperiodical: Introducing hexboard – a LaTeX package for drawing games of Hex. Hex is a strategic game with two players (Red and Blue) taking turns placing a stone of their color onto any empty space. A player wins when they successfully connect their sides together…

  • Boolean and Heyting islands

    Raymond Smullyan‘s logic puzzles frequently involve Knights (who always tell the truth) and Knaves (who always lie). In his book Logical Labyrinths (really a first course in propositional logic) he introduced islands where the lying or truth-telling habits can vary from day to day—that is, an inhabitant might lie on some days and tell the…

  • the bongcloud attack

    In this neverending pandemic there’s a shortage of stories putting a lasting smile on my face. Here’s one. If you are at all interested in chess, you’ll know by now that some days ago IGMs (that is, international grandmasters for the rest of you) Magnus Carlsen and Hikaru Nakamura opened an official game with a…

  • Penrose’s aperiodic tilings

    Around 1975 Sir Roger Penrose discovered his aperiodic P2 tilings of the plane, using only two puzzle pieces: Kites (K) and Darts (D) The inner angles of these pieces are all multiples of $36^o = \tfrac{180^o}{5}$, the short edges have length $1$, and the long edges have length $\tau = \tfrac{1+\sqrt{5}}{2}$, the golden ratio. These…

  • The strange logic of subways

    “A subway is just a hole in the ground, and that hole is a maze.” “The map is the last vestige of the old system. If you can’t read the map, you can’t use the subway.” Eddie Jabbour in Can he get there from here? (NYT) Sometimes, lines between adjacent stations can be uni-directional (as…

  • Heyting Smullyanesque problems

    Raymond Smullyan brought Knights and Knaves puzzles to a high art in his books. Here’s the setting: On Smullyan’s islands there are Knights, who always tell true statements, Knaves, who always lie, and sometimes also Normals, who sometimes tell the truth and sometimes lie. (image credit MikeKlein) Problems of this sort can be solved by…

  • Knights and Knaves, the Heyting way

    (image credit: Joe Blitzstein via Twitter) Smullyan’s Knights and Knaves problems are classics. On an island all inhabitants are either Knights (who only tell true things) and Knaves (who always lie). You have to determine their nature from a few statements. Here’s a very simple problem: “Abercrombie met just two inhabitants, A and B. A…

  • a SNORTgo endgame

    SNORT, invented by Simon NORTon is a map-coloring game, similar to COL. Only, this time, neighbours may not be coloured differently. SNORTgo, similar to COLgo, is SNORT played with go-stones on a go-board. That is, adjacent stones must have the same colour. SNORT is a ‘hot’ game, meaning that each player is eager to move…