Jamshrivelson

Question: The Robot Weightlifting World Championship’s final round is about to begin! Three robots, seeded 1, 2, and 3, remain in contention. They take turns from the 3rd seed to the 1st seed publicly declaring exactly how much weight (any nonnegative real number) they will attempt to lift, and no robot can choose exactly the same amount as a previous robot. Once the three weights have been announced, the robots attempt their lifts, and the robot that successfully lifts the most weight is the winner. If all robots fail, they just repeat the same lift amounts until at least one succeeds. Assume the following: all the robots have the same probability $p(w)$ of successfully lifting a given weight $w$; $p(w)$ is exactly known by all competitors, continuous, strictly decreasing as the w increases, $p(0) = 1,$ and $p(w) \rightarrow 0$ as $w \rightarrow \infty$; and all competitors want to maximize their chance of winning the RWWC. If $w$ is the amount of weight the 3rd seed should request, find $p(w).$ Give your answer to an accuracy of six decimal places. [Solution]

Question: Players A, B, C, and D are playing a game which starts with A saying the number $1.$ From that point on, each player says a number between $1$ and $4$ greater than the number spoken by the preceding player. The player who says $20$ wins the round, the player after them is removed from the game, the player after them starts the next round. Any other players survive to the next round. Each player’s top aim is to win the whole game but, if they realize that’s impossible, they’ll prioritize surviving to the next round. Who will survive the $4$ players game? [Solution]

Question: it’s the end of a close game of cornhole, and your team is $3$ points away from the win. By house rules, you have to hit the $3$ points exactly, or you lose. On your team are three athletes. The first is The Aggressor who has a $40%$ chance to get the cornhole, a $30%$ chance to hit the board and a $30%$ chance to miss entirely. The second is The Conservative who has a $10%$ chance to get the cornhole, a whopping $80%$ chance to hit the board, and a $10%$ chance to miss entirely. Finally, there’s The Waste who’s blind drunk at every game and will always miss the board. Given your roster and the free choice to use any of them in any situation, and you play optimally, what is the chance that you win the game? [Solution]

Question: it’s election night and your candidate is behind in the count. However, a significant fraction of the vote is still out in uncounted mail-in ballots. What are the chances that your candidates come back for the stunning victory if a whole bunch of people vote? What happens when the polls are tilted in one direction? [Solution]

Question: you’re watching the Moon from your room, alone, like every night. Seeking a higher purpose, you decide to track the projected area of the illuminated portion of the Moon throughout the month. If your data is accurate, how much faster will the illuminated Moon’s area be growing at half Moon as compared to a crescent Moon of $1/6^\text{th}$ area? [Solution]

Question: You’re playing a game of three on three basketball in a weird league. The rule on foul shots on a three pointer is that all three players from the team get to take turns shooting three foul shots. However, to get the second foul shot the team must make the first, and to get the second foul shot the team must make the second. As soon as a player misses, the foul shooting is over. If the players on your team have foul shot percentages $a,$ $b,$ and $c$ (no two equal), then what is the maximum number of shooting orders that would produce the same expected number of points? [Solution]

Question: in one of the great numerological miracles of our time, baseball players are obtaining batting averages with the same first three significant digits as the number of games in the stretch that the batting statistics are taken over. Is this an effect of rare planetary alignment? Suppose that a baseball player gets $4$ at bats in every of the $N$ games they play. What is the greatest value of $N$ for which it is impossible for the digits of their batting average (rounded to the nearest thousandth) to equal the number of games they’ve played in. [Solution]

Question: a regular staircase is built from blocks and the blocks in each level are different colors. The staircase can be built in whatever order that’s physically possible. However, whenever you make a choice, the Universal wave function splits so that each staircase actually exists in its own branch of the multiverse. In addition, the surface the stairs are built on is slightly sloped (in all verses), so any placed block slides forward until it hits the wall. How many universes will spawn to accommodate the possibilities for the $4$-level staircase? An $n$-level staircase? [Solution]

Question: relaxing at home for the $55^\text{th}$ week running, you have an idea to play reverse Jenga, stacking blocks one at a time to see how high you can get them before some blocks fall down. To make it a surprise, you place each new blocks center randomly at some place between the edges of the block underneath. About how many blocks do you expect to place before seeing a unit of blocks tumble down? [Solution]

Question: it’s now one year into quarantine and you’re well and truly out of ideas for fun on a Saturday night. To avoid falling asleep, you set up three pegs with three rings, each wider than the next, stacked widest to narrowest on the first peg. You want to move this stack to either of the other two pegs, moving one disk at a time, such that you never put a wider disk on top of a narrower disk. You know this is hard, but you have nothing but time. Suppose that instead of analyzing the situation, you simply move the disks at random, choosing uniformly from among the available valid moves at each step. How long should you expect to be at this exciting new amusement? [Solution]