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【正文】 llows us to pute Tn for any n we like. But nobody really like to pute from a recurrence， when n is large； it takes too long. The recurrence only gives indirect, local information. A solution to the recurrence would make us much happier. That is, we39。 because our construction proves only that 2Tn—1+1 moves suffice。s change our perspective and try to think big； how can we transfer a large tower? Experiments with three disks show that the winning idea is to transfer the top two disks to the middle peg, then move the third, then bring the other two onto it. This gives us a clue for transferring n disks in general： We first transfer the n?1 smallest to a different peg (requiring Tn1 moves), then move the largest (requiring one move), and finally transfer the n?1 smallest back onto the largest (requiring another Tn1 moves). Thus we can transfer n disks (for n 0)in at most 2Tn1+1 moves： Tn ≤2Tn—1+1， for n 0. This formula uses 39。s advantageous to LOOK AT SMALL CASES first. It39。s not immediately obvious that the puzzle has a solution, but a little thought (or having seen the problem before) convinces us that it does. Now the question arises： What39。 and their solutions all use the idea of recurrence, in which the solution to each problem depends on the solutions to smaller instances of the same problem. THE TOWER OF HANOI Let’s look first at a neat little puzzle called the Tower of Hanoi,invented by the French mathematician Edouard Lucas in 1883. We are given a tower of eight disks, initially stacked in decreasing size on one of three pegs: The objective is to transfer the entire tower to one of the other pegs, moving only one disk at a time and never moving a larger one onto a smaller. Lucas furnished his toy with a romantic legend about a much larger Tower of Brahma, which supposedly has 64 disks of pure gold resting on three diamond needles. At the beginning of time, he said, God placed these golden disks on the first needle and ordained that a group of priests should transfer them to the third, according to the rules above. The priests reportedly work day and night at their task. When they finish, the Tower will crumble and the world will end. 武漢科技大學(xué)本科畢業(yè)論文外文翻譯 2 It39。ll see repeatedly in this book that it39。s rules. Then T1 is obviously 1 , and T2 = 3. We can also get another piece of data for free, by considering the smallest case of all： Clearly T0 = 0， because no moves at all are needed to transfer a tower of n = 0 disks! Smart mathematicians are not ashamed to th
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