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Which way is the bike going? Can you deduce the direction of travel of a bike by the shape of the tracks it leaves? Sherlock Holmes thought so, but the answer is somewhat subtle. https://tinyurl.com/yetz489e #apieceofhtepi #substack #math #maths
Random coprime numbers The probability that two random large integers have no factors in common is 6 divided by π squared, which is about 60.8%. https://tinyurl.com/55ed8wky #apieceofthepi #piday #math #maths #substack
Identifying bottlenecks in networks A complicated network often consists of highly connected regions that are linked to each other by bottlenecks. How can we identify bottlenecks computationally? #substack #math #maths
The mathematics of the game Waffle Waffle is a word puzzle game along the lines of Wordle. In order to solve a game efficiently, it helps to know something about abstract algebra and the symmetric group. https://apieceofthepi.substack.com/p/the-mat
Turning a triangle into a square There is a well-known way to cut an equilateral triangle into four polygonal pieces and slide the pieces back together to form a square. It is impossible to do this with three pieces. https://tinyurl.com/ye2995dy #
@substack Constructing space-filling curves A new combinatorial construction produces infinitely many examples of space-filling curves, which are curves that completely fill the unit square. https://tinyurl.com/hvu3w22e #substack #maths #math
The Ramanujan Machine Can an A.I. be programmed to come up with conjectures about continued fractions and mathematical constants, in the style of Srinivasa Ramanujan? (The answer is “yes”.) https://tinyurl.com/43kyh28p #math #maths #su
Chess and the number e The number e is famous as the base of the natural logarithm, but it also appears in discrete combinatorial contexts, including chess problems. https://tinyurl.com/446fzrf9 #substack #maths #math
On @substack Chutes, ladders, and Markov chains A board configuration of Snakes and Ladders can be either ultimately winnable, occasionally winnable, or unwinnable. There are more possible configurations than there are atoms in the universe, and m
A polynomial with Rubik’s cube symmetry The Rubik’s cube group is a Galois group over the rationals, which means there is a polynomial whose roots have the same symmetries as the cube. https://tinyurl.com/yn298m36 #substack #math #math
The Parks puzzle The Parks puzzle is a Sudoku-like game that has recently been proved to be NP-complete. https://apieceofthepi.substack.com/p/the-parks-puzzle #substack #math #maths
Penny graphs A penny graph is the contact graph of an arrangement of unit circles on the plane. What happens if we do the same thing on the surface of a torus? https://apieceofthepi.substack.com/p/penny-graphs #substack #math #maths
Misère Connect Four If you are playing Connect Four as Player 2, and you are deliberately trying to lose, then there is a simple strategy that will force this to happen. https://apieceofthepi.substack.com/p/misere-connect-four #substack #ma
How many triangles are there? There is a formula for the number of triangles that are produced when a larger triangle is partitioned using straight lines that pass through its corners. https://apieceofthepi.substack.com/p/how-many-triangles-are-the
Wherever you go, I win! In the penultimate position of a game, the next player to move is bound to lose unless the other player makes a mistake. The analysis of such positions can be mathematically interesting. https://tinyurl.com/mw73abb2 #substa
My latest post from A Piece of the Pi: Tilings and metallic means The golden ratio is the first in a series of irrational numbers known as metallic means. Some of these numbers are related to aperiodic tilings, in the same way that the golden ratio
My latest post on A Piece of the Pi: Solving a maze with two exits There is an algorithm to solve a maze with two exits that guarantees an equal probability of finding either exit, no matter where the exits are. https://apieceofthepi.substack.com/p
My latest post on A Piece of the Pi: Maximally even musical scales It is possible to quantify the extent to which the notes in a musical scale are evenly spaced, and there is a sense in which the major scale is the most even. ➡️ https://apieceofth
My latest post on A Piece of the Pi: Square-free words and tangrams A square-free word is a string of symbols that has no consecutively repeating block. Given three different types of symbol, we can make arbitrarily long square-free words. ➡️ https
My latest post on A Piece of the Pi: Tredoku Patterns Tredoku is a generalization of Sudoku in which each 3 by 3 box forms part of the surface of a 3-dimensional stack of cubes. What are the possible shapes of a Tredoku board? ➡️ https://apieceofth