Jamshrivelson

Question: your friend Duane’s friend’s no-good, duplicitous, self-promoting grandchild has made extraordinary claims regarding their performance in the well-known cabin and road trip cardgame known as War. According to them, they’ve triumphed in a round where they won every single matchup with no ties, ending the game in a mere $26$ hands. Never one to let the feats of grandchildren go unchallenged, you set out to compute the probability of this occurrence. About how many rounds of War would one need to play to bear witness to their flawless victory? [Solution]

Question: Quality control at your ruler factory has taken a turn for the worst and your latest shipment of rulers are all broken into 4 pieces! What is the average length of the shard that contains the $\text{6 inch}$ mark? [Solution]

Question: It’s year 5 of quarantine and all computing power is reserved for face filters that make you look like a baby or the GEICO lizard. If you want to crunch numbers there’s a new calculator in town — the C. elegans petri dish. To add two numbers $x$ and $y$ you gather as many nematodes, put them in the dish, and come back the next day to see how many nematodes there are. The nematodes will pair bond (sex doesn’t matter, C. elegans are almost all hermaphroditic) and each pair will procreate (yielding one new worm), or not, with probability $1/2.$ It’s noisy, and it’s random, but that’s the best we can do in these trying times. To raise a number $x$ to the $n^\text{th}$ power, you gather as many nematodes and leave them in the dish for $n$ days, and however many there are upon your return is $x^n.$ Under these rules, what is $(1+1)^n$? [Solution]

Question: A gaggle of graduates gathers round a gargoyle who genuflects until they gear up in a circle upon which they (the gargoyle) announce, I thought no two of you were now in the same position relative to one another but alas I am mistaken, you comprise a number larger than $100$ such that you are the least populous class size for which it is simply not possible for less than $2$ of you to have remained in the correct relative position. How many students are in this graduation class? [Solution]

Question: Tortoise and Hare are having a stroll down a 10 mile road — Tortoise can run at a healthy clip of $60$ mph while Hare can run at an even healthier clip of $75$ mph. But that’s not all… the road is magic and every minute, on the minute, the road is stretched uniformly by $10$ miles. If Tortoise and Hare want to finish their stroll at the same time, how long should Hare hang back after Tortoise gets started? [Solution]

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]