Jamshrivelson

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]

Question: You’re in your town’s heat to head marble racing championship, the traditional way to determine who is the town’s next mayor. The race is split into two heats and your time in either heat is a random, normally distributed variable. If you have the fastest time in the first run, what is the probability $P_\text{win it all}$ that you end up winning the event, as determined by the sum of your times on heat run? Extra credit: what if there are $29$ other candidates in the race? [Solution]

Question: each night, you like to wind down with a relaxing game of one-upsmanship against yourself, throwing darts at your bullseye one by one, trying to get each one closer than the last. When a dart lands further from the center than the one that came before, the streak is over. Over the course of your life, how many darts will you throw on the average night? Assume that all darts hit inside the bullseye and that the darts are equally likely to land at any point inside the outer ring of the dartboard. For additional credit, you can play the demented version of the game where the board is divided into $10$ annuli, and instead of trying to move closer, period, you try to get within tighter and tighter rings. [Solution]

Question: Trapped in his house under Stage $4$ lockdown, Zach sits at the kitchen table with a half full glass and a cardboard disc, upon which is written the password he uses to edit the Riddler. Suddenly, he hears someone talking about cranberry sauce in the other room. He just can’t get enough of that cranberry sauce! When he gets up, he places the disc on the rim of the glass such that the disc’s center is at a random location on the inside of the rim. What is the probability that the cardboard disc falls into the half full glass of water, dissolving the password and forcing the Riddler into a $2$-week hiatus, because that is the only explanation for why Zach would abandon us like this? [Solution]

Question: another day, another demented hostage situation whose only salvation is twisted hat logic. The potentate of puzzles has kidnapped, blindfolded, and placed a red, green, or blue cap upon the head of you and $4$ of your unluckiest friends. Furthermore, they’ve split you up into two rows of $3$ and $2$ apiece. On opening your eyes, you can see the colors of the hats of the people in the opposite row. With nothing more than this information, and knowledge of your own position in the arrangement, you have to guess the color of your hat. If at least one person guesses correctly you all survive, otherwise it’s time for the long nap. Is there a strategy that guarantees your survival? [Solution]

Question: it’s Thanksgiving and your family is gathered ‘round the circular dinner table, Tofurkey in the middle, as is tradition. When the time comes, your Aunt Riddla brings out her famous cranberry sauce, handing it to you to place on the table. Wherever you place it, the person sitting there will take some sauce and then pass it randomly to one of their neighbors with probability $r$ to the right, and probability $\ell$ to the left. This placement is no small decision though. You want to punish your naughty Uncle Zach so that he gets the cranberry sauce last! Where should you start the sauce off on its journey if you want Uncle Zach to be the most likely to be the last person to get that famous cranberry sauce? [Solution]

Question: the hometown coin flipping team has staked out a healthy lead, in fact there’s a $\gt 99\%$ chance they bring home the championship (chip). Suddenly, disaster strikes, and they experience a total collapse and lose the game. If the game is a best of $101$ flips, what’s the chance to witness this devastating upset? [Solution]