Quant puzzles, solved step by step

The next term is 312211. Each term describes the one before it, read aloud as runs of equal digits. 1 is one 1, written 11. 11 is two 1s, written 21. 21 is one 2 and one 1, written 1211. 111221 is three 1s, two 2s and one 1, written 312211. The sequence is known as look-and-say.

Why does no arithmetic rule fit?

Try the usual moves first, as you would with any number series: differences, ratios, squares. 11 minus 1 is 10 and 21 minus 11 is 10, but 1211 minus 21 is 1,190, and the pattern dies. When the gaps jump like that, stop treating the terms as quantities. These terms are strings of digits, not numbers, and the rule works on the digits one run at a time. Think of reading a phone number to someone on a bad line: you say double two, triple five. Each term here is the previous one said that way and then written down.

How do you produce the next term without slipping?

Mark the runs first, then speak each run as a count followed by its digit. The slip people make is merging two runs of the same digit that have something else between them. In 111221 the three 1s at the front and the single 1 at the end are separate runs, so the reading ends with one 1, not four 1s. Bracket the runs on paper or in your head and read left to right: three 1s, two 2s, one 1 gives 312211. One more turn for practice: 312211 reads one 3, one 1, two 2s, two 1s, which writes as 13112221.

What can you say about the sequence beyond the next term?

Two facts show you understand the rule rather than just ran it. First, starting from 1, no digit above 3 ever appears. Neighbouring runs always hold different digits, so four equal digits can never sit in a row, no run is longer than three, and no count above 3 is ever written. Second, the terms grow at a steady rate: the 40th term has 63,138 digits, and each term is about 1.30 times as long as the one before, a ratio John Conway studied. Run the rule in a short loop to see both, which is also the coding follow-up many firms ask next.

On a timed screen, the point of a question like this is speed at dropping a wrong frame. Candidates who spend a minute hunting for a formula lose the minute; candidates who ask what else the digits could be doing find the rule in seconds.

36 passengers started. Work backwards from 16 and undo each step with its inverse. Growing by a third multiplies by 4/3, so undo it by multiplying by 3/4: 16 becomes 12. Undo half getting off by doubling: 24. Then 3/4 again: 18. Double again: 36. Forwards it checks: 36, 18, 24, 12, 16.

Why work backwards instead of setting up an equation?

Retracing your route to find a dropped wallet works because you know where you ended up. Here you know the final count and every step, so the cheapest route is to run the film in reverse. Each step is a multiplication, so each can be undone by multiplying by its reciprocal, starting from the 16 and moving toward the depot. An equation also works, 4x/9 = 16, but the backward chain shows every intermediate count, which lets you check that each one is a whole number of people.

What is the inverse of growing by a third?

This is the step that catches people. Growing by a third means the new count is 4/3 of the old one. To undo a rise of a third you multiply by 3/4, which removes a quarter of the new number, not a third of it. Taking a third off 16 gives 10.67, which is not a whole person and is a red flag on its own. Multiplying by 3/4 gives 12, and 12 grown by a third is 16 again, so the step checks.

The whole-number check also tells you what starting counts are possible at all. Every intermediate count, x/2, 2x/3, x/3 and 4x/9, must be a whole number, so x must be a multiple of 18. A free consistency check is worth saying aloud: 36 is a multiple of 18, and every count on the way, 18, 24 and 12, is whole. The same structure appears on a desk whenever a number passes through several percentage changes: a price up 10% and then down 10% ends at 99% of where it began, and undoing a change always means dividing by the factor, never subtracting the percentage.

200 miles. The car needs 100 / 50 = 2 hours to reach B, and the bird flies the whole time at 100 mph, so it covers 2 x 100 = 200 miles. Summing the zigzags gives the same answer: the first round trip is 133.3 miles, each later one is a third of the one before, and 133.3 / (1 - 1/3) = 200.

What is the question really asking you to count?

Think of a dog running back and forth between you and your front door while you walk home. You could trace every dash, or you could notice that the dog runs at a steady speed for exactly as long as your walk takes. Distance is speed times time, and the bird's flying time is fixed by the car, not by the zigzags, so the zigzag detail is a distraction. The car covers 100 miles at 50 mph in 2 hours; the bird flies at 100 mph for those same 2 hours. That is 200 miles, and it takes one sentence.

How do you check it by summing the zigzags?

The bird and car close the first 100 miles at a combined 150 mph, so they meet after 40 minutes, 33.3 miles from A. The bird flies back to B, 66.7 miles, arriving at 80 minutes, by which time the car is at 66.7 miles. Each round trip starts with the gap to the car one third of the previous gap, so the round trips form a geometric series with ratio 1/3. The first is 133.3 miles; the sum is 133.3 / (1 - 1/3) = 200. It agrees, and it shows why infinitely many turns still add to a finite distance.

Why do interviewers still ask a puzzle this well known?

Because the way you answer tells them more than the answer. A candidate who starts summing legs has reached for the first method that fits; a candidate who asks what quantity is fixed has found the invariant, and that is the habit the interviewer is hiring. The story about von Neumann summing the series in his head is part of the folklore; you get more credit for the one-line method and the series as a check. The same move, looking for a quantity that does not depend on the messy path, solves many expected-value and stopping questions on this page.