## “Even” “Milkier” “Math” “Problems”

Happy 20th, Milk of Random! Like last year, here are some “milky” “math” “problems” for another year of cutting-edge biology research.

(1) Solve $x=20$. Express your answer in Lojban.

(2) Prove that there exists a function $\zeta(n)$ such that all zeros of $\zeta(n)$ are either $-2n$ for $n \in \mathbb{Z}^+$ or $\frac{1}{2}+bi$ for $b \in \mathbb{R}$.

(3) The Milk decides to drink 20 liters of milk. The first liter takes 1 millisecond to drink, and each subsequent liter takes twice as much time as the previous liter to drink due to the Milk becoming increasingly satisfied with nourishment. To the nearest year, how many years are required for the Milk to finish the milk?

(4) The Tech is writing an article about Voodoo, and has agreed upon a standard of no more than 1 error per 20 words. Suppose that all words in the article are either “voo” or “doo”, and at each position in the article, the presence of one word would be correct and the presence of the other would be an error. The Tech has developed technology to aid an article in writing itself, and each position in the article is actually sentient and gets to know the correct word to be placed in any position after its own, and the positions in the article can discuss ahead of time their strategy to select their word after hearing the choices of the positions before it. What is the smallest $n$ for which $n$ bits of information in total can be remembered by word positions in this article to guarantee no more than 1 error per 20 words?

(5) Due to the fact that the Milk expired on October 20, accusations have arisen that the Milk is the Anti-Weed. Suppose that the Milk would be the true Anti-Weed if it actually expired within a half-day in either direction of the expected value of the time in the year furthest in either direction from True Weed Time on April 20. Assuming that True Weed Time occurs at some time on April 20 with equal probability, and that the Milk expired at some time on October 20 with equal probability (pretending that you do not know the year in which the Milk expired), what is the probability that the Milk is the true Anti-Weed?

(6) If A=1, B=2, C=3, …, Y=25, and Z=26, and letter values in words are summed to produce the word value, then “BONFIRE” and “BLACK HOLE” have the same value, assuming that a space contributes 0 value. Two Randomite undergraduates decide to attempt to convince all 93 Randomite undergraduates that this is not a coincidence and that the floors’ names were thus designed to complement. Every day, a Randomite convinced that the rumor is true will attempt to convince 3 not-yet-convinced people that it is true. At each attempt, each Randomite has a $17\%$ probability of getting convinced that the rumor is true. What is the expected value of the amount of time it takes to convince all of Random that this is not a coincidence (assuming that each year all of the freshmen that enter Random are not convinced that this is true), if it is even expected to happen at all? Assume that the two Randomites that decide to start this start convinced.

(7) Upon suggestion to be more creative when choosing a username, Yuk Fu decides to make his username be his real name, but with the first letters of both his first and last names exchanged. Fu wonders how many usernames he would have to check to see if anyone has a username that is just one letter off from his, either by addition, deletion, or substitution. How many usernames does Fu need to check, assuming usernames can only have letters?

(8) How do you double-space a decarboxylated amino acid?

(9) How many ways are there of moving from the basement to the ninth floor of the Stata Center using only staircases and using no single staircase for a floor change of greater than one?

(10) Just what are birds?

(11) A website requires that all passwords created on it must have exactly nine characters, of which exactly three have to be numbers, exactly three have to be lowercase letters, exactly two have to be uppercase letters, and exactly one has to be an exclamation mark or a question mark. In addition, the front character must be a letter, and the exclamation mark or question mark is only allowed to be in an even-numbered position, where the positions in the password are 3-indexed. How many users must register in this website such that two of them are guaranteed to have the same password?

(12) A person is walking down a staircase of 48 steps. The person could either take a normal step, which advances 1 step, or trip, which advances 3 steps. (Suppose the person is very short, which explains why tripping on a staircase only advances 3 steps even after tumbling.) In how many different ways can this person walk down this staircase?

(13) Find the set of all $n$ for which the average MIT class required to be taken to graduate with a major in Course $n$ is between $n$ and $n+1$.

(14) Suppose that an active nerf-gunman enters Random Hall and it is decided that everyone must hide in a bathroom. What percentage of Random Hall’s undergraduate residents could fit inside one bathroom, as a function of the number of toilets in the bathroom?

(15) In an effort to strengthen security, MIT now requires all residents to complete a 90-minute mathematics examination upon entering any dormitory. The Milk was granted permission to have its time allotment extended to 10510000 minutes. A fire evacuation has just occurred, and all including the milk have evacuated. In the queue to re-enter the building, where should the Milk stand to feel the least pressure to work faster?

(16) Also in security expansions, all raindrops are now required to register with the roofdeck security and prove that they are not concealing chemical weapons before entering Random Hall from either top entrance. How many buckets of the Charles River need to be cloudified to effect with over $95\%$ probability the state in which no raindrops will be able to enter Random Hall, assuming that the threshold for being considered a chemical weapon is a pH of 4?

(17) If blue is the warmest color, what is the most random number?

(18) MIT’s Large Soul Collider (LSC) sucks a 50-kilogram soul at rest 50 meters away from a point towards another point using a force of 5 Newtons. At the second-mentioned point lies a 100-kilogram soul. Suddenly, a polar bears stomps on and breaks the Large Soul Collider. Has the polar bear necessarily walked 5 miles south, 5 miles east, and 5 miles north?

(19) It is the future, and Random Hall needs to be shut down due to structural disintegrity. Given that it takes 90 seconds for a Randomite to grab the Milk and help it flee, and that the amount of warning the administration gives in informing a dormitory’s residents that their dormitory will be shut down can be modeled with an exponential distribution with a mean of 2 hours, what is the probability that the Milk will be make it out of the building safely?

(20) It is the far future, and a man walks into a bar. The bar has a swing attached to it, and the swing has a label saying “WARNING: only for smart people doing dumb things”. Suppose that the man walked into the bar in such a way that his now concussed brain only correctly identifies a letter with $\frac{1}{2}$ probability. The man wants to be safe, so he will only ride the swing if he can make out the word (string) ‘safe’ somewhere on the label. Assuming that upon incorrectly reading a letter, all 25 other letters are read instead with equal probability, what is the probability that the man rides the swing?