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The Magic of Mathematics
The Mi-Mu Game
An exploration of Axiomatic Systems
Aidan Sproat
Games
What is a game?
• Starting position
• Rules
• End position
The Mi-Mu Game
There are three objects in the game: the letters M, I and U. They
may be combined to form words. Any string of letters is a word.
All valid words begin with the letter M and are followed by a
combination of the letters I and U.
The initial word is MI. The aim is to transform it into MU by
applying the following rules:
1. Add a U to the end of any string ending in I.
2. Double the string after the M (that is, change Mx, to Mxx).
3. Replace any III with a U.
4. Remove any UU.
Chess problem - 1
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Your task: To formulate the following chess puzzle as a game,
and to determine whether the game is solvable.
You need: A start position, some rules and an end position.
The problem: move a knight from one corner of a chessboard
to an adjacent corner.
Chess problem - 2
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Your task: To formulate the following chess puzzle as a game,
and to determine whether the game is solvable.
You need: A start position, some rules and an end position.
The problem: move a bishop from one corner of a chessboard
to an adjacent corner.
Maths is a formal system
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Axioms
Logic
Conjectures
Theorems
Proof
Maths is a game
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Starting positions – axioms, problems
Rules – logic, theorems
Finishing positions – conjectures, solutions
Each game – proof
A teeny-weeny bit of algebra
• The golden rule of algebra:
If you apply the same function to two equal quantities,
you obtain the same result
• In easier terms:
Do the same thing to both sides and everything will be
ok.
Revisiting the Mi-Mu Game
Transform MI to MU using the following rules:
1. Add a U to the end of any string ending in I.
2. Double the string after the M (that is, change Mx,
to Mxx).
3. Replace any III with a U.
4. Remove any UU.
Tabulate your steps.
Forming and proving conjectures
• Add two columns to your table, recording the
number of Is and Us present in each answer.
• List your observations. If you need to gather more
information, do. If you spot any patterns, see if you
can do anything to break the patterns.
• When you find a pattern that you think holds, try
to work out whether it is useful to solve our
problem.
• Can you use your understanding of the rules to
prove that your conjecture will always be true?
8
7
6
5
4
3
2
1
A
B
C
D
E
F
G
H
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