Chapter 5: Q18E (page 370)
Prove that Algorithm 1 for computing n! when n is a nonnegative integer is correct.
The required algorithm is proved by induction.
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Prove that the first player has a winning strategy for the game of Chomp, introduced in Example 12 in Section 1.8, if the initial board is two squares wide, that is, a board. [Hint: Use strong induction. The first move of the first player should be to Chomp the cookie in the bottom row at the far right.]
Prove that if and B are sets, then
Let a be an integer and d be a positive integer. Show that the integers qand r with and which were shown to exist in Example 5, are unique.
Use strong induction to show that all dominoes fall in an infinite arrangement of dominoes if you know that the first three dominoes fall, and that when a domino falls, the domino three farther down in the arrangement also falls.
a) Find a formula for
by examining the values of this expression for small
values of n.
b) Prove the formula you conjectured in part (a).
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