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]

How much can you pull out of a hat?

This week’s Riddler Classic is a strategy game about maximizing payout. What is the optimal strategy?

You start with just the number 1 written on a slip of paper in a hat. Initially, there are no other slips of paper in the hat. You will draw from the hat 100 times, and each time you draw, you have a choice: If the number on the slip of paper you draw is k, then you can either receive k dollars or add k higher numbers to the hat.

For example, if the hat were to contain slips with the numbers 1 through 6 and you drew a 4, you could either receive $4 or receive no money but add four more slips numbered 7, 8, 9 and 10 into the hat. In either case, the slip with the number 4 would then be returned to the hat.

If you play this game perfectly — that is, to maximize the total amount of money you’ll receive after all 100 rounds — how much money would you expect to receive on average?

My solution:
[Show Solution]

Optimal Wordle

This week’s Riddler Classic is about the viral word game Wordle.

Find a strategy that maximizes your probability of winning Wordle in at most three guesses.

Here is my solution:
[Show Solution]

Outthink the Sphinx

This week’s Riddler Classic is a tricky puzzle that combines logic and game theory.

You will be asked four seemingly arbitrary true-or-false questions by the Sphinx on a topic about which you know absolutely nothing. Before the first question is asked, you have exactly $1. For each question, you can bet any non-negative amount of money that you will answer correctly. That is, you can bet any real number (including fractions of pennies) between zero and the current amount of money you have. After each of your answers, the Sphinx reveals the correct answer. If you are right, you gain the amount of money you bet; if you are wrong, you lose the money you bet.

However, there’s a catch. (Isn’t there always, with the Sphinx?) The answer will never be the same for three questions in a row.

With this information in hand, what is the maximum amount of money you can be sure that you’ll win, no matter what the answers wind up being?

Extra credit: This riddle can be generalized so that the Sphinx asks N questions, such that the answer is never the same for Q questions in a row. What are your maximum guaranteed winnings in terms of N and Q?

If you’re just looking for the answer, here it is:
[Show Solution]

Here is a more detailed write-up of the solution:
[Show Solution]

Flipping your way to victory

This week’s Riddler Classic concerns a paradoxical coin-flipping game:

You have two fair coins, labeled A and B. When you flip coin A, you get 1 point if it comes up heads, but you lose 1 point if it comes up tails. Coin B is worth twice as much — when you flip coin B, you get 2 points if it comes up heads, but you lose 2 points if it comes up tails.

To play the game, you make a total of 100 flips. For each flip, you can choose either coin, and you know the outcomes of all the previous flips. In order to win, you must finish with a positive total score. In your eyes, finishing with 2 points is just as good as finishing with 200 points — any positive score is a win. (By the same token, finishing with 0 or −2 points is just as bad as finishing with −200 points.)

If you optimize your strategy, what percentage of games will you win? (Remember, one game consists of 100 coin flips.)

Extra credit: What if coin A isn’t fair (but coin B is still fair)? That is, if coin A comes up heads with probability p and you optimize your strategy, what percentage of games will you win?

Here is my solution:
[Show Solution]

Optimal HORSE

This week’s Riddler Classic is about how to optimally play HORSE — the playground shot-making game. Here is the problem.

Two players have taken to the basketball court for a friendly game of HORSE. The game is played according to its typical playground rules, but here’s how it works, if you’ve never had the pleasure: Alice goes first, taking a shot from wherever she’d like. If the shot goes in, Bob is obligated to try to make the exact same shot. If he misses, he gets the letter H and it’s Alice’s turn to shoot again from wherever she’d like. If he makes the first shot, he doesn’t get a letter but it’s again Alice’s turn to shoot from wherever she’d like. If Alice misses her first shot, she doesn’t get a letter but Bob gets to select any shot he’d like, in an effort to obligate Alice. Every missed obligated shot earns the player another letter in the sequence H-O-R-S-E, and the first player to spell HORSE loses.

Now, Alice and Bob are equally good shooters, and they are both perfectly aware of their skills. That is, they can each select fine-tuned shots such that they have any specific chance they’d like of going in. They could choose to take a 99 percent layup, for example, or a 50 percent midrange jumper, or a 2 percent half-court bomb.

If Alice and Bob are both perfect strategists, what type of shot should Alice take to begin the game?

What types of shots should each player take at each state of the game — a given set of letters and a given player’s turn?

Here is how I solved the problem:
[Show Solution]

and here is the solution:
[Show Solution]

Pool hall robots

This Riddler puzzle is about arranging pool balls using a robot!

You own a start-up, RoboRackers™, that makes robots that can rack pool balls. To operate the robot, you give it a template, such as the one shown below. (The template only recognizes the differences among stripes, solids and the eight ball. None of the other numbers matters.)


First, the robot randomly corrals all of the balls into the wooden triangle. From there, the robot can either swap the location of two balls or rotate the entire rack 120 degrees in either direction. The robot continues performing these operations until the balls’ formation matches the template, and it always uses the fewest number of operations possible to do so.

Using the template given above — a correct rack for a standard game of eight-ball — what is the maximum number of operations the robot would perform? What starting position would yield this? How about the average number of operations?

Extra credit: What is the maximum number of operations the robot would perform using any template? Which template and starting position would yield this?

Here is my solution:
[Show Solution]

The number line game

This Riddler puzzle is a game theory problem: how should each player play the game to maximize their own winnings?

Ariel, Beatrice and Cassandra — three brilliant game theorists — were bored at a game theory conference (shocking, we know) and devised the following game to pass the time. They drew a number line and placed \$1 on the 1, \$2 on the 2, \$3 on the 3 and so on to \$10 on the 10.

Each player has a personalized token. They take turns — Ariel first, Beatrice second and Cassandra third — placing their tokens on one of the money stacks (only one token is allowed per space). Once the tokens are all placed, each player gets to take every stack that her token is on or is closest to. If a stack is midway between two tokens, the players split that cash.

How will this game play out? How much is it worth to go first?

A grab bag of extra credits: What if the game were played not on a number line but on a clock, with values of \$1 to \$12? What if Desdemona, Eleanor and so on joined the original game? What if the tokens could be placed anywhere on the number line, not just the stacks?

Here are the details of how I approached the problem:
[Show Solution]

And here are the answers:
[Show Solution]

Pick a card!

This Riddler puzzle is about a card game where the goal is to find the largest card.

From a shuffled deck of 100 cards that are numbered 1 to 100, you are dealt 10 cards face down. You turn the cards over one by one. After each card, you must decide whether to end the game. If you end the game on the highest card in the hand you were dealt, you win; otherwise, you lose.

What is the strategy that optimizes your chances of winning? How does the strategy change as the sizes of the deck and the hand are changed?

Here is my solution:
[Show Solution]