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Consider an infinite sequence of positions 1, 2, 3, . . . and suppose we have...

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Advance Math

Consider an infinite sequence of positions 1, 2, 3, . . . andsuppose we have a stone at position 1 and another stone at position2. In each step, we choose one of the stones and move it accordingto the following rule: Say we decide to move the stone at positioni; if the other stone is not at any of the positions i + 1, i + 2,. . . , 2i, then it goes to 2i, otherwise it goes to 2i + 1.

For example, in the first step, if we move the stone at position1, it will go to 3 and if we move the stone at position 2 it willgo to 4. Note that, no matter how we move the stones, they willnever be at the same position.

Use induction to prove that, for any givenpositive integer n, it is possible to move one of the stones toposition n. For example, if n = 7 first we move the stone atposition 1 to 3. Then, we move the stone at position 2 to 5Finally, we move the stone at position 3 to 7.

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