Jamshrivelson

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]

Question: it’s your last chance to be on Jeopardy and you want to leave it all on the field. You’ve trained for years and are confident that you can answer every question correctly and beat your competitors to the buzzer every time. Your only issue is how to order your guesses. The Daily Double tile is placed randomly on the board, so you can’t predict its location. The two strategies you’ve narrowed it down to are guessing all the tiles in succession (all $\$200$ tiles then all $\$400$ tiles and so on) or to guess tiles randomly. What is the expected value of each strategy, assuming you have the whole “getting the answers right” part covered? [Solution]

Question: after a life of flipping fair coins, you’ve had enough. Your new mission is to construct elaborate and worthless schemes to extract virtual fair coins out of $3$ flips of a crooked one. Ever the positivist, you want to know just how many crooked coins there can be that suit your campaign. For how many values of $p$ can a fair coin be simulated using no more than $3$ flips? [Solution]

Question: you’re practicing for the forgotten NYC marathon. On the first day you set your treadmill to $9$ minutes per mile. On the second day, you set it to steadily “speed up” over time from $r_0 = 10$ minutes per mile at the beginning down to $r_1 = 8$ minutes per mile at the end (when the run is half over, the treadmill moves at $9$ minutes per mile). Which run took less time? [Solution]

Question: It’s Halloween Eve and famous ambassador (ambassador of pumpkins) David S. Pumpkins has appeared out of the clear blue sky to invade your apartment complex’s elevator. “Join me”, he entices, for a few rounds of “Damn That’s a Hot Pumpkin” his crazy new game that’s been sweeping the nation. Altogether, there are $61$ people, including David S. Pumpkins himself. The rules are simple, everyone gets in a circle, including David S. Pumpkins, and starts counting from $1$ to a number $N$ that’s specified by David S. Pumpkins himself. When his watch strikes “pumpkin time” the players start passing the pumpkin to the left, counting the numbers one by one, until they get to $N.$ When this happens, the person who said $N$ is eliminated, and the next round begins, starting with the person to their left. Distrusting of his pumpkin ways, you hang back and watch a few rounds from the wall. You see that the first person to be eliminated was $18$ positions to the left of David S. Pumpkins, the second person was $31$ positions to the left of whoever started their round, and the third person eliminated was the very person who began round three! What is the smallest value of $N$ for which this would be possible? After solving that, David S. Pumpkins proffers another: who will win this game? After solving that, David S. Pumpkins has had it up to here with other people winning his hot pumpkin games, and says “I’ll show you how to do it!!1” Putting aside the constraints from the first two problems, what value of $N$ should David S. Pumpkins pick so that he himself is the person to win the game Damn That’s a Hot Pumpkin? [Solution]

Question: It’s the day after the championship parade and you are Lebron James. Doc Rivers has been deposed, Paul George’s name is on the trade winds, and Patrick Beverley hasn’t thought about trash talk in over a month. Needless to say, life is good. But one day, your brother in arms, The Brow, Anthony Davis himself, challenges you to some $1$ on $1,$ first to score wins! Due to his dominant height and wingspan, he’s able to grab every single rebound, even the ones off his own three pointers! “What’s a king to do?”, you wonder. But then you realize — your career steal line is a full $0.2$ steals per game higher than Davis’. Every possession, you have the chance to steal the ball away with probability $L_\text{steal}.$ If yours and Anthony’s shot percentages are $L_\text{score}$ and $D_\text{score},$ respectively, then what is the chance that you, Lebron James, win the $1$ on $1$ game? Inversely, how high does your steal probability, $L_\text{steal},$ need to be for it to be an even match? [Solution]