CHALLENGEGRADE 9-12 / Age 14+The crossing puzzle......... A board consists of twenty squares arranged in a rectangle with four rows of five (see figure 1). On squares 1, 6, 11, and 16, there are white pieces, and on squares 5, 10, 15, and 20, there are black pieces placed. With each move, one of the 8 pieces is shifted diagonally along an arbitrary number of squares (to a free square); hence, if you choose the piece placed on square 11, then you can move it to any of the squares 17, 7, or 3. By moves of this type, you have to reach a final position in which the white pieces are on squares 5, 10, 15, 20 and the black pieces on squares 1, 6, 11, 16; thus the pieces must cross the board. The problems is to achieve this in the smallest possible number of moves.
Figure 1 The Weight Puzzle of Bachet (1587-1638)... To weigh an object with a pair of scales or a balance, we need a set of weights. For weighing to an accuracy of one gram, the set usually consists of the following weights (in grams): 1-1-2-5-10-10-20-50-100-100-200-500-1000; hence, two weights of 1 g, one weight of 2 g, one weight of 5 g, two weights of 10 g, etc. We then speak of a standard set of weights. This provides weights with which we can make all combinations from 1 g up to and including 1999 g. This leads us to pose the problem: For a given number of weights, determine the set of weights which enable us to weigh the largest possible number of objects whose weights increase by single grams (starting with 1 g).
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