7/23/2019 invariant game

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Invariants & Games

Tony Liu, IMSA Math Circle

Thursday, October 12, 2006

So, what are invariants? Simply put, they are things that don’t change. They don’t vary. You’re all

probably too familiar with games (insert laughter here), but you’re probably wondering how invariants canever be useful. Consider the following problem:

Example 1  Suppose we have a regular  8× 8  chessboard and we remove two opposite corners (so we are left with a mutilated board with  8×8−2 = 62 squares). Is it possible to tile this board with non-overlapping  1×2dominoes? 

Let’s put this more formally. Imagine a system on which you can perform various operations. You’d liketo understand and analyze the behavior of the system, to determine what positions can be reached from whatother positions. Invariants are often useful for this purpose. The idea of a system with a set of operationsnaturally lends itself to the topic of games. Below is potpourri of these types of problems. Enjoy!

Problems

1. There are 1 white, 2 black, and 3 red chips on a table. In one step, you may choose two chips of different colors and replace them with a chip of the third color. Can we ever end up with just onechip? What if we started out with 1 white, 3 black, and 3 red chips?

2. Two players alternately color squares of a 4× 4 chessboard. The loser is the one who first completesa 2× 2 subsquare. Who wins?

3. A dragon has 100 heads. A knight can cut off 15, 17, 20, or 5 heads with one blow of his sword. In eachof these cases, 24, 2, 14, or 17 new heads, respectively, grow on its shoulders. If all heads are blown off,the dragon dies. Can the dragon ever die?

4. Two players take turns drawing diagonals of a regular octagon. They may connect two vertices if thediagonal draw does not intersect with an earlier one. The first person who cannot make a move loses.Who wins? Try replacing the octagon with a regular 2006-gon.

5. The numbers 1, 2, 3, . . . , 10 are written on a blackboard. You are allowed to erase any two numbers  a

and  b  and replace them with  a + b− 1. After 9 of these operations, what number will be left?

6. A beetle sits on each square of a 5 × 5 chessboard. At a signal, each beetle crawls diagonally onto aneighboring square. Then it may happen that serveral beetles will sit on some squares and none onothers. Find the minimal possible number of free squares.

Harder Problems

1. You have three piles of stones containing 5, 49, and 51 stones. You can join any two piles together intoone pile and you can divide a pile with an even number of stones into two piles of equal size. Can youever achieve 105 piles each with one stone?

2. (USSR ’89) A 23× 23 square is completely tiled by 1 × 1, 2× 2, and 3× 3 tiles. What is the minimumnumber of 1× 1 tiles needed?

3. (USAMO ’99) The Y2K Game is played on a 1×2000 grid as follows. Two players in turn write eitheran S or an O in an empty square. The first player who produces three consecutive boxes that spell SOSwins. If all boxes are filled without producing SOS then the game is a draw. Prove that the secondplayer has a winning strategy.

4. Can we pack 250 1 × 1× 4 bricks into a 10× 10× 10 box?

That’s it for today. Tune in next week for another exciting episode of Math Circle!

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