Month: May 2023

We played a two player strategy game today, using all the information from +plus magazine.

The game is played as follows. At the start of a game each player decides on their three colour sequence for the whole game. The cards are then turned over one at a time and placed in a line, until one of the chosen triples appears. The winning player takes the upturned cards, having won that “trick”. The game continues with the rest of the unused cards, with players collecting tricks as their triples come up, until all the cards in the pack have been used. The winner of the game is the player that has won the most tricks. An average game will consist of around 7 “tricks”.

e.g. Player 1 picks RRB, Player 2 picks RBB and see who wins.

The question we thought about was:

• Is it just a game of chance, or if you are choosing second, could you improve your chance of winning?

There is a related game called Penny’s game, using Heads and Tails of a coin toss instead of playing cards. In this version, you just play until one person has won a “trick”. We worked out some of the odds given in the first table in this article.

We ended up having to sum a geometric series! Great fun!

This game is thanks to David Bedford at the BCME Conference in Warwick.

One person writes down a polynomial with positive integer coefficients. Call it f(x) They then choose an integer that it bigger than all their coefficients. Call it n. They then calculate f(n) and give just n and f(n) to the second person.

The second person should be able to name that polynomial! How?

Example:

Given only n=8 and f(n) = 6855 how could you work out that the polynomial was

?

It might help to think about an example when n=10 first.

Here is a cheatsheet for when you have a strategy. Can you write some Python code to work out the polynomial for you?

We recreated Ben Sparks geogebra app which demonstrates how seeds would look being sowed at different fractions of a full turn.

Here is some polar graph paper to draw patterns by hand first.

You might like to try with 1/4, 1/5, 3/8, 5/9 and moving out by one circle each time you sow a seed.

On Geogebra you can set up a slider r and then use Sequence((0.5 i; i r*2 π),i,1,150)

Lights Out was an electronic game released by Tiger Electronics in 1995. The game consists of a 5 by 5 grid of lights. When the game starts, a random number or a stored pattern of these lights is switched on. Pressing any of the lights will toggle it and the adjacent lights. The goal of the puzzle is to switch all the lights off, preferably with as few button presses as possible.

Here, or below, you can play a 3 x 3 version on Geogebra (courtesy of Stephen Jull and Yin Su).

One method to solve – Light Chasing

“Light chasing” is a method similar to Gaussian elimination which always solves the puzzle (if a solution exists), although with the possibility of many redundant steps. In this approach, rows are manipulated one at a time starting with the top row. All the lights are disabled in the row by toggling the adjacent lights in the row directly below. The same method is then used on the consecutive rows up to the last one. The last row is solved separately, depending on its active lights.

Here is a worksheet showing all the possibilities (effectively) for the last row. Can you solve them all, and hence solve any 3×3 grid?

Here are links to play the 4×4 game and the 5×5 game.