Betting on football with future knowledge

This week’s Fiddler is a football-themed puzzle with a twist: you can see the future! Sort of.

You know ahead of time that your football team will win 8 of their 12 remaining games, but you don’t know which ones. You can place bets on every game, placing bets either for or against your team. You can bet any amount up to how much you currently have. You want to implement a betting strategy that guarantees you’ll have as much money as possible after the 12 games are complete. If you did so, then after the 12 games how much money would you be guaranteed to have if you started with $100?

My solution:
[Show Solution]

A more elegant alternative solution, due in large part to a clever observation by Vince Vatter.
[Show Solution]

Optimal baseball lineup

This week’s Fiddler is a problem about how to set the optimal baseball lineup.

Eight of your nine batters are “pure contact” hitters. One-third of the time, each of them gets a single, advancing any runners already on base by exactly one base. (The only way to score is with a single with a runner on 3rd). The other two-thirds of the time, they record an out, and no runners advance to the next base. Your ninth batter is the slugger. One-tenth of the time, he hits a home run. But the remaining nine-tenths of the time, he strikes out. Your goal is to score as many runs as possible, on average, in the first inning. Where in your lineup (first, second, third, etc.) should you place your home run slugger?

Extra Credit: Instead of scoring as many runs as possible in the first inning, you now want to score as many runs as possible over the course of nine innings. What’s more, instead of just having one home run slugger, you now have two sluggers in your lineup. The other seven batters remain pure contact hitters. Where in the lineup should you place your two sluggers to maximize the average number of runs scored over nine innings?

My solution:
[Show Solution]

Braille puzzle

This week’s Fiddler is a counting problem about the Braille system.

Braille characters are formed by raised dots arranged in a braille cell, a three-by-two array. With six locations for dots, each of which is raised or unraised, there are 26, or 64, potential braille characters.

Each of the 26 letters of the basic Latin alphabet (shown below) has its own distinct arrangement of dots in the braille cell. What’s more, while some arrangements of raised dots are rotations or reflections of each other (e.g., E and I, R and W, etc.), no two letters are translations of each other. For example, only one letter (A) consists of a single dot – if there were a second such letter, A and this letter would be translations of each other.

Of the 64 total potential braille characters, how many are in the largest set such that no two characters consist of raised dot patterns that are translations of each other?

Extra Credit In addition to six-dot braille, there’s also an eight-dot version. But what if there were even more dots? Let’s quadruple the challenge by doubling the size of the cell in each dimension. Consider a six-by-four array, where a raised dot could appear at each location in the array. Of the 224 total potential characters, how many are in the largest set such that no two characters are translations of each other?

My solution:
[Show Solution]

Making something out of nothing

This week’s Fiddler is a problem about composing functions. Here it goes:

Consider $f(n) = 2n+1$ and $g(n) = 4n$. It’s possible to produce different whole numbers by applying combinations of $f$ and $g$ to $0$. How many whole numbers between $1$ and $1024$ (including $1$ and $1024$) can you produce by applying some combination of $f$’s and $g$’s to the number $0$?

Extra Credit: Now consider the functions $g(n) = 4n$ and $h(n) = 1−2n$. How many integers between $-1024$ and $1024$ (including $-1024$ and $1024$) can you produce by applying some combination of $g$’s and $h$’s to the number $0$?

My solution:
[Show Solution]

Möbius prism

This week’s Fiddler is a problem about a multi-sided Möbius prism:

Consider an three-dimensional prism whose bases are regular N-gons. I twist it and stretch it into a loop, before finally connecting the two bases. Suppose that my twist is by a random angle, such that the two bases are aligned when they are coonected. Among all whole number values of N less than or equal to 1,000, for which value of N will a randomly twisted regular N-gon prism have the most distinct faces, on average?

My solution:
[Show Solution]