1 acrostic twins a move is to turn over two coins

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1. Acrostic Twins. A move is to turn over two coins, either in the same row or the same column, with the southeast coin going from heads to tails. The Sprague-Grundy function satisfies g ( x, y ) = mex { g ( x, b ) , g ( a, y ) : 0 b < y, 0 a < x } , (4) this being the least number not appearing earlier in the same row or column as ( x, y ). If y = 0, this game is just nim, so g ( x, 0) = x ; similarly g (0 , y ) = y , the function being symmetric. It is easy to construct the following table of the values of g ( x, y ); each entry is just the mex of the numbers above it and to the left. When we do we find a pleasant surprise — that g ( x, y ) is just the nim-sum, x y . 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 0 3 2 5 4 7 6 9 8 2 2 3 0 1 6 7 4 5 10 11 3 3 2 1 0 7 6 5 4 11 10 4 4 5 6 7 0 1 2 3 12 13 5 5 4 7 6 1 0 3 2 13 12 6 6 7 4 5 2 3 0 1 14 15 7 7 6 5 4 3 2 1 0 15 14 8 8 9 10 11 12 13 14 15 0 1 9 9 8 11 10 13 12 15 14 1 0 Table 5.1. For Acrostic Twins, g ( x, y ) = x y . I – 32
To fix the ideas, consider the game with layout T T T T T T T T H T T T T T T T T H T T T T T T H (5) We use the convention when referring to a position at ( x, y ) in such a layout, that x represents row and y represents column, where rows and columns are numbered starting at 0. Thus, the three heads are at positions (1,3), (3,2) and (4,4). The heads are at positions with SG-values, 2, 1 and 0. Since 2 1 0 = 3, this is an N-position. It can be moved to a P-position by moving the SG-value 2 to SG-value 1. This is done by turning over the coins at positions (1,3) and (1,0). (There are other winning moves. Can you find them?) 2. Turning Corners. Let us examine a more typical game. A move consists of turning over four distinct coins at the corners of a rectangle, i.e. ( a, b ), ( a, y ), ( x, b ) and ( x, y ), where 0 a < x and 0 b < y . The Sprague-Grundy function of this game satisfies the condition g ( x, y ) = mex { g ( x, b ) g ( a, y ) g ( a, b ) : 0 a < x, 0 b < y } . (6) Since we require that the four corners of the selected rectangle be distinct, no coin along the edge of the board may be used as the southeast coin. Therefore, g ( x, 0) = g (0 , y ) = 0 for all x and y . Moreover, g (1 , 1) = 1 and g ( x, 1) = mex { g ( a, 1) : 0 a < x } = x by induction (or simply by noticing that the game starting at ( x, 1) is just nim). The rest of Table 5.2 may be constructed using equation (6). As an illustration, let us compute g (4 , 4) using equation (6). The are 16 moves from position (4,4), but using symmetry we can reduce the number to 10, say ( x, y ) for 0 x y 3. Computing the SG-values of each of these moves separately, we find g (4 , 4) = mex { 0 , 4 , 8 , 12 , 1 , 14 , 11 , 3 , 5 , 2 } = 6. If (5) represents a position in Turning Corners, we can find the SG-value easily as 3 1 6 = 4. This is therefore an N-position, that can be moved to a P-position by moving the SG-value 6 to an SG-value 2. There is a unique way that this may be done, namely by turning over the four coins at (3,3), (3,4), (4,3) and (4,4). T T T T T T T T H T T T T T T T T H T T T T T T H −→ T T T T T T T T H T T T T T T T T H H H T T T H T 5.3 Nim Multiplication. The entries in Table 5.2 may seem rather haphazard, but such an appearance is deceptive. In fact, the function g defined by (6) may be considered I – 33
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 0 6 11 13

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