FIRST ANSWERS Any positive odd integer can be expressed as 2n + 1, where n is any non-negative integer. Substituting this ex pression for a in the above formula gives us: 25(2n + l)2 + l
2
which simplifies to 50(n2 + n) + 13. This is the total cost of the magazines in dollars where n is any positive integer. Since n2 + n is always an even number, the total cost must be a multiple of 100 dollars with 13 dollars added to it. Therefore, to raise the total to a multiple of 100, the gratuity is always 100 - 13 = 87 dollars regardless of my wife’s age or how many magazines I bought! When I got home I discovered that the magazines were not in as good condition as I thought. One was missing pages 8, 9, 13, 27, 28, and 33. How many leaves had been tom from it?
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M M m m g
Observe that the figure has five compartments. If there is a solution, the twisty line that enters a compartment from the outside, then leaves the compartment, must cross two line segments—one while entering, one while leaving. If the compartment is surrounded by four line segments, as in the case of compartments A and B , the line can enter and leave the compartment twice. However, if a compartment is sur rounded by five line segments, all five can be crossed only if the line has one of its ends inside the compartment. Three compartments (C, D, and E) are each surrounded by five line segments. Therefore, if the puzzle is solvable, each of these compartments must contain one end of the twisty line. But a line has only two ends. Therefore, there is no way to solve the puzzle without leaving at least one line segment uncrossed. Couth left his daughter alone to work on the proof, then
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