# Mathematical Games

Some of these games have worked well with 12 year olds. First the teacher takes on a pupil. Then pupils play against each other in pairs. Then pupils who think they've "got it" try against the teacher.

## NIM

Players take turns to remove any number of objects from one of n rows. The player who takes the last object wins.

Ans:If a game is played with a finite set of numbers and each move changes only one set and the game terminates, it is equivalent to NIM.

To determine who should win, write the sizes of the piles in binary. Add the columns (no carrying). If the sum of each column is even, the person to play loses. (from Mathematical Carnival)

In the reverse game the player who takes the last counter loses
Ans: The strategy in the reverse game follows the normal game until only one row has more than one counter. Then you take away either all the biggest row or all but one of this row so as to leave an odd number of rows.

Variations include
• PRIM - Remove from a heap of n any prime
• DIM - Remove a divisor of n from a heap of n
• Remove any square number from a heap.
• RIMS - Draw some dots. Players take turns to join them up. No loops are allowed to overlap. Dots within topological regions give a NIM game.
• Sprouts - Draw a few (3 is enough) dots on a piece of paper. Players take turns to join one dot to another or itself, then put a new dot on the line. A line can't cross itself or another line, nor can it pass through a dot. No spot can have more than 3 lines going to it. The winner is the person who makes the last move. (from Mathematical Carnival)
• Rayles - same as RIMS except that only 1 or 2 spots can be joined.
• Kayles (isomorphic to Rayles) Skittles are lined up. Can knock down two adjacent skittles, or one skittle. The last player wins.
• Grundy - Split any heap into 2 non-empty heaps.

## Noughts and Crosses

• In a variation of the game, where the first person to make a line loses, the first player can only force a draw if they first play in the centre (in "Mathematical puzzles & diversions")
• Make 9 cards with the digits 1-9 on them (one per card). Scatter the cards face-up. Players take turns picking a card. The 1st to have 3 cards which add up to exactly 15 wins.
Ans: this game is isomorphic to Noughts and Crosses where the cells have the following values
```2 9 4
7 5 3
6 1 8
```
• Make 9 cards with the following words on them (one per card) HOT FORM WOES TANK HEAR WASP TIED BRIM SHIP. Scatter the cards face-up. Players take turns picking a card. The 1st to hold 3 cards with same letter on wins.
Ans: this game is isomorphic to Noughts and Crosses where the cells have the following values
```HOT  FORM WOES
TANK HEAR WASP
TIED BRIM SHIP
```

## Grid Games

1. Draw a 3x3 grid. Players can color in any amount of squares on one row or column. The winner is the one who fills in the last square.
Ans: The 2nd player can always win. If the first player fills a single cell in, then the 2nd player should fill 2 cells in to make an 'L'. Otherwise the 2nd player should complete an 'L' or 'T' consisting of 5 cells.
2. TacTix - Draw a 4x4 grid. Players can color in any amount of adjoining squares on one row or column. The winner is the one who fills in the last square. Ans: Gardner in "Mathematical puzzles & diversions" says that the second player can always win, but there's no simple strategy. The reverse game is much harder. A variation, where the colored in squares needn't be adjoining, is harder still.

## Miscellaneous

1. 10x10
• 2 Dice. Shade in a rectangle of size X by Y, where X and Y are die scores. If X=Y, the player can shade in X*Y squares anywhere (and needn't finish exactly)
• 1 die. Fill in a rectangle with the area.
2. 2 players take turns to add 1-10 to a common total. First to 100 wins.
Ans: (First to 89 wins)
3. Tug of war
```    | | | |X| | | |
```
Each player has 50 muscle points to gamble. At each turn the one who gambles the most pulls the knot one square their way. The player who pulls the knot to their end wins.
4. 13 petals (Sam Lloyd) Remove 1 or 2 connected petals. The one who takes the last wins.
Ans: (preserve symmetry)

## References

1. On Games and Numbers, J.H. Conway
2. Mathematical Carnival, Martin Gardner, Penguin 1976
3. Mathematical puzzles & diversions, M. Gardner, London, 1961.
4. Mathematical Games, C. Lukacs and E. Tarjan, Souvenir, 1969