by Mike Slattery

This document describes a version of a game invented by Lewis Carroll in 1872. The rules given here have been somewhat simplified to make computer implementation easier.

Based on a description of the game in *The Diaries of Lewis Carroll*
edited by R. Green (1954).

The actual game of Croquet is played on a lawn by hitting balls across the ground with mallets. The players construct a course beforehand consisting of wire hoops through which each player needs to hit their ball in order. The first player to hit their ball through the hoops in the proper order and then hit the wooden peg at the end of the course is the winner. The game Arithmetical Croquet was invented by Carroll as a mental game which one could easily play without any equipment (e.g. sitting on a beach or taking a walk). It is loosely based on an imaginary croquet course laid out in a straight line along the number line. There is a hoop at each multiple of ten (10, 20, ...) and the winning peg at 100. The position of each player's imaginary ball is represented by a number and play proceeds from left to right increasing along the number line toward 100.

**Rules**

1. The first player names a (positive whole) number not greater than 8: the second does the same: the first then names a higher number, not advancing more than 8 beyond his last; and so on alternately - whoever names 100, which is the 'winning peg', wins the game.

2. The numbers 10, 20, etc. are the 'hoops'. To 'take' a hoop, it is necessary to go, from a number below it, to one the same distance above it: e.g. to go from 17 to 23 would 'take' the hoop 20: but to go to any other number above 20 would 'miss it' and would be considered an illegal move.

3. It is also lawful to 'take' a hoop by playing 'into' it, in one turn, and out of it, to the same distance above it in the next turn: e.g. to play from 17 to 20, and then from 20 to 23 in the next turn, would 'take' the hoop 20. A player 'in' a hoop may not play out of it with any other than the number so ordered.

4. Whatever step one player takes, bars the other from taking an equal step, or the difference between it and 9: e.g. if one player advances 2, the other may not advance 2 or 7. But a player has no 'barring' power when 'in' a hoop, or when at any number between 90 and 100, unless the other player is also at such a number.

5. The 'winning peg' may be 'missed' once. Once past 100, play proceeds in the opposite direction with the player decreasing his number according to the same rules as above. Missing the 'peg' twice loses the game.

6. When one player is 'in' a hoop, the other can keep him in, by playing
the number he needs for coming out, so as to bar him from using it.
He can also do it by playing the difference between this and 9. And
he may thus go on playing the 2 barring numbers alternately: but he may
not play either twice running: e.g. if one player has gone from 17 to 20,
the other can keep him in by playing 3, 6, 3, 6, etc.