## Thursday, July 9, 2009

### The Game of Take Away

Directions
To play, begin with a set of 16 pennies or other small objects. Two players take turns removing pennies. At each turn a player must remove anywhere from 1 to 4 pennies (inclusive). The winner will be the last player who makes a legal move.

In the classroom, allow students to play for awhile. Ask them to look for patterns and come up with strategies without telling other classmates. When you feel that some students have ideas, allow them to discuss what they have noticed with the class. See where this takes you!! Have fun!

Explanation of the Game and How It Works
Of course the winner takes the last 4-1 pieces and wins the game by leaving zero pieces.
In order to do this the winner must present their opponent with exactly five pieces. The opponent can take anywhere from one to four pieces leaving you with a win.
So are there more options than leaving five pieces? Yes! All multiples of five could be presented.
This idea can be shown as a list of oasis and deserts. Because the winner must leave zero pieces, an oasis ≡ 0 (mod 5). Remember that a player can move from desert to oasis (winning spot), but never from oasis to oasis. In other words, the winner always wants to present their opponent with an oasis amount of pieces, leaving themselves in a desert.

0 1 2 3 4 5 6 7 8 9 10 11 12 . . .

Now that students understand the idea of oasis and deserts make the game more difficult by changing the rules. What if a player can only remove 2-4 pieces or 3-4 pieces. What will the oasis and deserts look like?

Hint: Removing 2-4 pieces OASIS ≡ 0, 1 (mod 6)
Removing 3-4 pieces OASIS ≡ 0, 1, 2 (mod 7)

Change the rules around and encourage your students to develop new strategies. Take Away Poison requires the winner to leave only one piece. You can also change the number of pieces at the beginning of the game as well as the number of pieces that can be removed on each turn to change the game.