Jamshrivelson

Question: Mira the baby architect has a tapering post and some rings. If placed alone, each ring will “find its height” and rest there. If a ring with a higher rest height is placed before one with a lower rest height, the higher one will cut the bottom of the post off from ring placement. How many different ways are there to stack a ring tower? [Solution]

Question: It’s 8:59 at the $N$-lane town pool and the coronavirus is absolutely ripping. In an effort to mitigate the spread, swimmers must stay at least one lane apart. In one minute, the swimmers will hop into the pool, one lane at a time, until it becomes impossible to obey social distancing. If there are $N$ swimmers, how many do you expect to be left crying on the side of the pool? [Solution]

Question: Suppose you have a pencil, a hexagon, and absolutely nothing to do. What is the greatest number of knockoff hexagons you can draw using the same $6$ points as the original hexagons? What about septagons? What about octagons? What about … [Solution]

Question: King Auric’s collection of unique golden spheres of integer radius $\left(1\text{ cm}, 2\text{ cm}, \ldots\right)$ is the object of his covetous children’s eyes. To stave off fratri- and regicide, he will divide the gold up evenly by weight and bequeath an equal share unto each. He has the minimum number needed to do this. How many spheres does he have if he has $3$ children? What if he has $C = \{2,3,4,5,6,\ldots\}$ children? [Solution]

Question: A bacterial colony starts from a single cell that has a probability $\gamma$ of splitting into $2$ and a probability $\left(1-\gamma\right)$ of lysing, as do all of its descendants. The strain in question is Riddlerium classicum, about which not much is known apart from its cataclysmic reproductive viability, $\gamma = 80\%$ (real bacteria are far more successful, Fig 4 in Robust Growth of Escherichia coli). What’s the probability that the colony is blessed with everlasting propagation? (i.e. the population never crashes to zero) [Solution]

Question: You’re making a sign using a marker to draw the letters. The marker tip is a circle that’s $2\text{ cm}$ across. If you want the marks to be as uniform as possible (as measured by the standard deviation of the ink intensity), and you can’t place the tip within $1\text{ cm}$ of where it’s previously been, how far apart should you make the marks? [Solution]

Question: In one of the greatest disasters in the history of collective planning, all 330M residents of the U.S. join the same Zoom call by picking two random times between 8:00 and 9:00. They dial in at the earlier time and hang up at the later time. What is the probability that there is at least one person who is on for some portion of everyone else’s call, a so-called Superzoomer? Further, what is the chance of there being two Superzoomers on the same call? [Solution]

Question: Find the longest words that share letters with only 49 of the 50 U.S. states. Extra: find the state with the most such words. [Solution]

Question: A post can accumulate spam comments, and each spam comment can accumulate spam comments. If each open reply thread accumulates spam at a rate of $1$ reply per day, then what is the expected number of spam posts by the end of a three day weekend? [Solution]

Question: You can make 100 coin flips with either of two coins, Coin A worth $\pm1,$ and Coin B worth $\pm2.$ You survive if you end up with a positive score. How should you flip the coins to maximize your chances? [Solution]