Puzzle Collection

This is a list of puzzles I have found interesting at various points in my life.

Puzzle 1

There is a square table with a coin on each corner. The table is behind a curtain and hidden from you. Your goal is to get all the coins to be heads up: at any point, if you have achieved the goal, you will be told about it, and win. The only way to affect the coins is to tell the person behind the curtain to flip as many coins as you like, in specified corners: a sample instruction might look like: ‘flip the coins in the top right and bottom left corners.’ However, after every flip you orchestrate, the table is spun and ends up in random configuration you do not know.

Design a series of moves that guarantees a win in a finite number of moves.

Challenge: do the same for a $ 3 \times 3 $ grid of coins spun about its centre. Can you do this for a table shaped as an n-gon, with n coins?

Puzzle 2

You have an infinite number of Jenga blocks: rigid blocks with length $L$ and mass $m$. You are placing them over the edge of a table to build a tower, such that you maximize the overhang of these blocks. How can you maximize the overhang, and what is the maximum overhang of $n$ blocks? What happens as you keep adding more and more blocks?

Puzzle 3

Tom starts at $(0,d)$, and Jerry starts at $(0,0)$, the origin. Jerry runs with constant velocity $u$ along the positive x-axis, and Tom runs with constant speed $u$, always in the direction towards Jerry at that instant. Provide a condition for Tom to successfully catch Jerry, and, given this condition is true, the time it takes for Tom to catch Jerry.

Puzzle 4

You flip a coin until you get three heads in a row. What is the expected number of times you flip this coin?

Puzzle 5

A basketball player has a free-throw average of $70\%$ at some point in the season, and some time later, has a free-throw average of $80\%$. Is it necessarily true that it was exactly $75\%$ at some point?

Puzzle 6

We will play an extremely simplified version of poker, with a deck of three cards of an Ace, King, and Queen, and only two players. Both players initially put in a dollar into the pot, and then each draw a card. The first player then has the choice to bet an additional dollar into the pot, to check (end the turn with no action), or to fold. The second player can choose to either call (match the money the first player has put into the pot), or to fold. If the second player calls, then we go to a showdown, and the player with the bigger card (Ace beats King beats Queen) wins the entire pot. What is the optimal strategy for each player, and which player would you rather be?

Puzzle 7

Buckle up, we’re going meta. There is an ant walking along the number line, that originally starts at $2$. At each time step, it takes a step either to the left or the right with equal probability. The ant falls off a cliff at either $n=0$ or $n=10$. What is the probability it falls off at $n=0$ Try to figure out why the following ‘solution’ is incorrect. ‘Solution’: Consider the last time the ant has passed $n=2$ before falling off. Call it time $T$. At this point, it has a $\frac{1}{2}$ probability of going to either $n=1$ or $n=3$. Say it goes to $n=1$. Then, it cannot go to $n=2$, since it has already been there for the last time, and thus it must fall off at the next time step. A similar reasoning holds for if it steps to $n=3$. Thus, where the ant falls off from is completely determined by the move it makes at time $T$. Thus, it has an equal probability of falling off $n=0$ or $10$, and thus the answer is $\frac{1}{2}$