SIG Interview Guide

Without multiplying it out, work out how many decimal digits the number $100!$ has.

A lottery drum holds $84$ balls numbered $1$ through $84$. After a thorough mixing, five balls are drawn out one at a time. A ticket costs $\$1$ and pays $\$x$ if the five numbers emerge in strictly increasing or strictly decreasing order; otherwise it pays nothing. What value of $x$ makes the ticket exactly break-even in expectation?

A carnival game uses a board of $8 \times 8$ painted squares, each square measuring $2$ inches per side. A player flicks a circular token of diameter $1$ inch so that its center lands uniformly at random somewhere on the board. To win, the token must sit entirely inside one painted square — touching or crossing any painted gridline, or hanging off the board's outer edge, loses. What is the player's probability of winning?

A soldering robot travels across a circuit board from pad $(0,0)$ to pad $(7,4)$, moving one unit at a time either right $(R)$ or up $(U)$. Its firmware refuses to execute the same direction three times consecutively — at most two identical moves may occur in a row. How many move-sequences satisfy this constraint?

Shares of Helios Robotics will do exactly one of two things by tomorrow's open: double, or fall by half. Separately, a colleague is happy to take either side of an even-money (1:1) wager on which direction the stock moves. Suppose you put $A$ whole dollars into the stock and stake $B$ whole dollars on the wager, choosing sides so that your total profit comes out identical whether the stock jumps or drops. What are the smallest whole-dollar values of $A$ and $B$ that lock in a positive profit either way?

Cards numbered $1$ through $n$ lie face-up on a desk. You play against an auditor under one rule: you may claim card $x$ only while at least one proper divisor of $x$ is still on the desk. The instant you claim $x$, the auditor confiscates every proper divisor of $x$ that remains. Play stops when you have no legal claim left, and the auditor pockets everything still on the desk. For a general $n$, describe a strategy that makes your total as large as possible, and explain the reasoning behind it.

A street performer fans out four identical sealed envelopes; exactly one contains a $\$100$ note and the other three are empty. You tear envelopes open one at a time, paying a fee of $X$ dollars for each one you open, and you quit the moment the note appears. At what fee $X$ does neither side hold an edge — that is, the game has zero expected profit?

Five hundred point-sized beetles land at independent, uniformly random spots on a taut 1-foot wire. Each one immediately flips a fair coin to pick a direction and crawls at 1 foot per minute. Whenever two beetles meet head-on, both instantly turn around and continue crawling; a beetle that reaches either end of the wire drops off. How long, in expectation, until the wire is completely clear?

A game shop's bargain bin holds three fair dice: a three-sided die with faces 1–3, a four-sided die with faces 1–4, and an ordinary cube with faces 1–6. You grab one die uniformly at random and roll it once. Without listing individual outcomes, find the expected value of your roll.

A jeweler has $8$ gold rings that look identical, but exactly one contains extra metal and is a bit heavier than the rest. Using only a two-pan balance (no weights), how few weighings guarantee finding the heavy ring?

A weather station carries three redundant thermometers whose readings $X, Y, Z$ are independent and uniformly distributed on $[0, 3]$. To resist outliers, the station logs the median of the three readings. What is the probability that the logged value falls between $1$ and $2$?

An interviewer slides a sealed box across the table: a standard double-six domino set, every tile distinct. "Make me a market on the number of tiles inside — then tell me the exact fair value." A double-six set contains every possible tile whose two ends each show between $0$ and $6$ pips, with no tile repeated. How many tiles is that, and how should you quote before you've worked it out exactly?