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Chapter 1: Problem 31
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Fast Eddie needs to double his money; he can only do so by playing a certain win-lose game, in which the probability of winning is \(p\). However, he can play this game as many or as few times as he wishes, and in a particular game he can bet any desired fraction of his bankroll. The game pays even money (the odds are one-to-one). Assuming he follows an optimal strategy if one is available, what is the probability, as a function of \(p\), that Fast Eddie will succeed in doubling his money?
It is a standard result that the limit of the indeterminate form \(x^x\), as \(x\) approaches zero from above, is 1 . What is the limit of the repeated power \(x^{x^{. ^{. ^{. ^{x }}}}} \) with \(n\) occurrences of \(x\), as \(x\) approaches zero from above?
The proprietor of the Wohascum Puzzle, Game and Computer Den, a small and struggling but interesting enterprise in Wohascum Center, recently was trying to design a novel set of dice. An ordinary die, of course, is cubical, with each face showing one of the numbers \(1,2,3,4,5\), 6. Since each face borders on four other faces, each number is "surrounded" by four of the other numbers. The proprietor's plan was to have each die in the shape of a regular dodecahedron (with twelve pentagonal faces). Each of the numbers \(1,2,3,4,5,6\) would occur on two different faces and be "surrounded" both times by all five other numbers. Is this possible? If so, in how many essentially different ways can it be done? (Two ways are considered essentially the same if one can be obtained from the other by rotating the dodecahedron.)
Define a function \(f\) by \(f(x)=x^{1 / x^{x^{1 / x^{x^{1 / x ^{. ^{. ^{. }}}}}}}} (x>0)\). That is to say, for a fixed \(x\), let $$ a_1=x, \quad a_2=x^{1 / x}, \quad a_3=x^{1 / x^x}=x^{1 / x^{a_1}}, \quad a_4=x^{1 / x^{x^{1 / x}}}=x^{1 / x^{a_2}}, \quad \ldots $$ and, in general, \(a_{n+2}=x^{1 / x^{a_n}}\), and take \(f(x)=\lim _{n \rightarrow \infty} a_n\). a. Assuming that this limit exists, let \(M\) be the maximum value of \(f\) as \(x\) ranges over all positive real numbers. Evaluate \(M^M\). b. Prove that \(f(x)\) is well defined; that is, that the limit exists.
Suppose we are given an \(m\)-gon (polygon with \(m\) sides, and including the interior for our purposes) and an \(n\)-gon in the plane. Consider their intersection; assume this intersection is itself a polygon (other possibilities would include the intersection being empty or consisting of a line segment). a. If the \(m\)-gon and the \(n\)-gon are convex, what is the maximal number of sides their intersection can have? b. Is the result from (a) still correct if only one of the polygons is assumed to be convex? (Note: A subset of the plane is convex if for every two points of the subset, every point of the line segment between them is also in the subset. In particular, a polygon is convex if each of its interior angles is less than \(\left.180^{\circ}.\right)\)
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