3.3 Game of binary bits
3.3.3 Binary bits game
In the game of binary bits we start off with a m×n rectangular array of zeros and ones. There are m rows and n columns and only the numbers 0 and 1 can appear. As is often the case, the numbers are called bits. A legal move of the game is described this way:
1. The player selects a 1 bit in some position and changes it to a 0 bit. 2. The player may optionally change any or all of the bits in the column
below the selected bit 1.
Play evidently stops when all the bits have been changed to 0. The player who made the last legal move wins.
At first it seems obvious that the game eventually stops. A moment’s reflec- tion, however, may give us pause. As the game progresses some moves may add 1 bits, so the total count of 1 bits does not always go down. In Problem101
you are asked to show that the game is finite. This, we recall, is essential if our analysis in terms of balanced and unbalanced positions is to be successful. Example 3.3.1 A move in a 5×3 game is illustrated in Figure 3.9 Here the player elected to change one of the 1 bits in the second column, and he also flipped two of the lower bits.
Can you spot whether this was a good move? Was there a better move? ◭
The m×1 game Here there is but one column and the strategy should be obvious. The player to start simply chooses the topmost 1 bit and changes that bit and all the ones below it to zero bits. The game is over and he wins. A position with any 1 bits is unbalanced.
1 1 0
1 0 0
0 0 0
0 1 0
0 1 1
0 1 1
0 1 1
0 0 1
0 1 0
0 1 0
Figure 3.9: A move in a 5×3 game of binary bits.
The m×2 game Here there are two columns and the strategy is obvious . . . after some thought. In Problem102 you are asked to solve the game. The strategy that works is called the mirror strategy and plays an important role in game theory.
The m×3 game Here there are three columns and the strategy is no longer obvious at all. At this point the game becomes rather more interesting. We know that an analysis of balanced and unbalanced positions will result in a completely solved game but we do not yet know how to do that in any simple way.
Problem 101 Show that every game of binary bits must end in a finite number
of steps. Answer
Problem 102 Find a complete strategy for the m×2 game of binary bits.
Answer
Problem 103 Which, if any, of the positions in the 5×3 games of Figure3.10
are balanced? Answer
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
Figure 3.10: Which positions are balanced?
Problem 104 Which, if any, of the positions in the 5×3 games of Figure3.11
3.3. GAME OF BINARY BITS 111 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0
Figure 3.11: Which positions are balanced?
1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0
Figure 3.12: Which positions are balanced?
Problem 105 Which, if any, of the positions in the 5×3 games of Figure3.12
are balanced? Answer
Problem 106 Do you have a conjecture? Answer
Problem 107 Define a position in a m×3 game to be even if there are an even number of 1 bits in each row. Define a position in a m×3 game to be odd if there is at least one row containing an odd number of 1 bits. Check each of the following:
1. The end position of the game is even.
2. If a player makes a move from an even position it will surely result in an odd position.
3. If a player faces an odd position there is always a choice of move that leaves an even position.
Answer
Problem 108 Give a complete solution for the m×3 game of binary bits.
Answer
Problem 109 Are you prepared to announce a solution for the m×n game of
Problem 110 Describe all the balancing moves in the coin game displayed in
Figure3.7. Answer
Problem 111 In the coin game one can change the rules to allow more coins
in each pile. For example:
1. Each play of the game requires a player to remove all the coins of the same type from one of the piles.
2. Optionally the player may also add coins to or subtract coins from the same pile provided the coins added or subtracted are of lower value than the coins initially removed. (E.g., a player removes all dimes and then can add or subtract as many pennies and nickels as he pleases, but cannot add any quarters.
3. The player to take the last coin is the winner of the game.
How does this change the game? Answer
Problem 112 In the coin game one can change the rules to allow any player to
keep the coins that he has removed. How does this change the game?
Answer
Problem 113 (A number game) A game similar to binary bits starts with a
m×n rectangular array of arbitrary numbers. A legal move of the game is to change any nonzero number to zero and, optionally, change any or all of the numbers in the column below the selected number. The last player to move
wins. Analyze this game. Answer
Problem 114 (A word game) This word game is also similar to the game of
binary bits. The players start with three or more words. A player moves in this game by selecting a word and a letter that appears in that word. He must remove all appearances of that letter in the word chosen and may, optionally, add in or remove any other letters that are earlier in the alphabet. For example if the six words are
[Twas brillig and the slithy toves]
then a legal move would be to select the “l” in brillig and remove both of them. The “r” cannot be removed but the other letters can and any letters a—k could be added in, for example
brillig abbrek
would be allowed. The last player to move wins. Analyze this game.
Answer
Problem 115 Are you prepared to announce a solution for the game of Nim?