Celebrating the playful magic of John Horton Conway

Hello. Ahead of the start advanced course “Mathematics for Data Science”, have prepared for you the translation of an article that was written in memory of the legendary mathematician John Horton Conway.

We invite you to entertain yourself by solving a number puzzle, a geometric puzzle, or playing a game with random patterns, inspired by the playful genius of the legendary mathematician.

Legendary mathematician John Horton Conway, who died in April from COVID-19, childishly sincerely fond of invention of puzzles and games… He carried out detailed analysis of many puzzles such as catfish cube, peg solitaire and Conway’s soldiers… He invented “doomsday algorithm(A quick method of calculating the day of the week in your head – Conway could do this in less than two seconds) and countless games, including game “Seedling” (Sprouts) and famous game “Life”, which marked the beginning of the study of cellular automata.

Most of Conway’s serious mathematical work also stems from his weakness for mathematical games. He made original contributions to group theory (Leach Bars, monstrous nonsense hypothesis), multidimensional geometry, tessellation (mosaic), knot theory, number theory (surreal numbers), algebra, mathematical logic and analysis.

This month we celebrate the playful genius of the famous British mathematician with two puzzles and an exploration game. First, we’ll play with Conway’s number puzzle, which, without undue modesty, is the epitome of perfection itself. Next, enjoy a geometric puzzle related to some of his most visually delivering pieces. Finally, we will dive into the open ended game created by the reader Quanta, which is reminiscent of Conway’s cult game Life.

Puzzle 1: Digital Excellence

There is a mysterious 10-digit decimal number abcdefghij… All his numbers are different, and they have the following properties:

a divisible by 1
ab divisible by 2
abc divisible by 3
abcd divisible by 4
abcde divisible by 5
abcdef divisible by 6
abcdefg divisible by 7
abcdefgh divisible by 8
abcdefghi divisible by 9
abcdefghij divisible by 10

What is this number?

Before you start solving this puzzle, take a moment to admire the absolute perfection of its shape. It is presented completely naturally, without the slightest arbitrariness or contrivance. After reading the first two conditions, you know exactly what the rest of the puzzle will be. And when this natural set of conditions results in a unique response, it’s amazing. For me, as a puzzle maker, this number substitution puzzle evokes the same feeling that Mozart inspired Einstein, who said that Mozart’s music “was so pure that it seemed to be always present in the universe, waiting to be discovered by a master. “Only a numerically gifted person like Conway could grasp such a perfect platonic form from the puzzle garden of Eden!

You can, of course, solve this puzzle by brute-force search with a computer, but this is not necessary. I encourage you to do this with pencil and paper. All of these type of digit substitution puzzles can be solved using a two-step process familiar to Sudoku users: first you establish relationships between numbers, which narrows the possibilities, and then you systematically search for unknown numbers through trial and error. Here, you should use the tricks you learned in school to determine if a number is divisible by a given digit. If you get the most out of the puzzle, you won’t be left with too many candidates to search through trial and error.

If you want to make it harder for yourself, try to solve this puzzle completely in your head. After all, Conway was known for solving math problems “with bare hands“. It takes a lot of attention and patience, but I assure you that it is possible.

Puzzle 2: Dual Triangles

There is an isosceles triangle that contains an angle equal to x degrees. The ratio of two sides of different lengths is y…

It turns out that not one, but two whole different triangles have the same values. x and y!

What are the meanings x and y for these two isosceles triangles? What is special about these triangles and how do they relate to Conway’s work?

For our final puzzle, you will need paper and pencil again. It is even better if you take a few sheets in a cage. This is a game that will immerse you in Conway’s way of thinking – you will draw small diagrams and create various structures, as he did in his games “Seedlings” and “Life”. Our game, which creates structures like polyomino, was introduced Quanta by a reader named Jona Raphael.

Puzzle 3: Random hairy patterns

You have an infinite plane on which you place square tiles. You must add new tiles in turn, randomly, so that each new tile has at least one edge in common with the previously placed tile. The probability of a tile being placed in any particular location is proportional to the number of edges of previously placed tiles that border that location.

Let’s look at two examples:

If you only have one tile so far, then the second tile has an equal chance of being north, south, east, or west of the original tile.

If you have a ring of eight tiles, then there are 12 positions around the outside of the ring and one position in the middle, all valid for placing the next tile. The middle one is four times more likely to get a tile than any outside eligible position, because it shares four edges with the previously placed tiles, not just one.

We rate “hairiness” (H, because the original uses “hairiness”) or “appearance” of any configuration as the number of open edges of the tiles divided by the number of tiles. For instance:

For one tile on a plane H = 4 edges ÷ 1 tile = 4.
For a ring of eight tiles H = 16 edges ÷ 8 tiles = 2.
For a row of eight tiles H = 18 edges ÷ 8 tiles = 2.25.

The reciprocal H, can be called “internals” or compact configuration.

The goal of this game is pure exploration. Unlike most of the games we are used to, excluding Conway’s Life, this is “endless game without a player “as Conway described his creation. For me, the dynamics of our play is reminiscent of how adult children balance their desire to stay close to their parents with their desire to act independently. Parameter H is a measure of how much they diverge while remaining in minimal contact, while its reciprocal, interior, is a measure of how close are they…

(In this 2014 video Numberphile(number of phil(John Conway explains how he made Life.)

Here are some questions that will guide your exploration of this square world.

A new tile can be placed next to one edge (touching only one other tile), in a corner (touching two), inside the letter “U” (touching three), or inside a hole (touching all four). How does each placement affect the number of open edges in a new configuration (pattern)?
What are the minimum and maximum values H and what types of tile patterns do they correspond to? Can you come up with an approximate or exact formula for the maximum and minimum values H as the number of tiles increases (n)?
What is the expected value H (approximate or accurate) for a given value n?
Find the smallest pattern that is “balanced” so that adding the next tile is as likely to increase the number of open edges as it decreases it. Can you find a symmetrical pattern that has this property?
Find the smallest pattern for which the expected value H remains unchanged after adding another tile. What is the next smallest pattern by the number of tiles that this property is valid for?

That’s all for a start. Now you can explore this game on your own. Try to find answers to some interesting new questions. Maybe you will find a new structure (please share pictures of interesting patterns!) Or even prove a theorem. And while you are doing this, suggest a name for this game.

Play – give Conway’s spirit a smile!