Tech companies try to keep their interview questions secret, so that prospective employees can’t prepare answers in advance. Applicants reclaim the power by sharing their interview experiences publicly on websites like Glassdoor and CareerCup. As a result, what were intended as forums to help career hopefuls navigate the job market have become like a WikiLeaks for puzzle lovers. This week, I’m declassifying Bezos’s documents and delivering them straight to your browser window, with free shipping, of course.
Our first puzzle comes from a Software Engineer interview, according to a post on CareerCup. It has a neat premise: labeling the faces on a blank die to achieve a given goal, but in case you solve it too quickly I’ve supplemented it with two bonus puzzles that also explore blank dice.
Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!
Puzzle #44: Blank Dice
Amazon interview question: If you roll two typical dice and sum their results, you’ll get a number between 2 and 12. However, not every number is equally likely (e.g. you’ll only get a 2 if you roll snake eyes, whereas 7s will come up often). Suppose you have one normal die and one blank die (a small cube with no numbers on it). Label the blank die with some subset of the numbers 0, 1, 2, 3, 4, 5, 6 so that when you roll both dice, all sums from 1 to 12 are equally likely. You must label every face of the blank die, and you may repeat numbers.
Bonus questions about blank dice, courtesy of Dmitry Kamenetsky: Given two blank dice A and B, label them with the digits 1 through 12 once each (no repeats) so that when you roll them, there is a 50% chance that A rolls higher than B and a 50% chance that B rolls higher than A.
Label three blank dice using the digits 1 through 18 once each (no repeats) so that when you roll them, each die has an equal chance of being the highest.
Although these might seem like a tedious exercise in trial and error, they all have pretty intuitive solutions. I found the last problem particularly satisfying, but you have to work up to it by solving the smaller case first.
I’ll be back Monday with the answers and a new puzzle. Do you know a cool puzzle that you think should be featured here? Message me on X @JackPMurtagh or email me at [email protected]
Solution to Puzzle #43: Anarchy Tic-tac-toe
Did you figure out how to crush me in last week’s tic-tac-toe variant? The first player can always force a win as follows. Shout-out to Kari Roth for emailing a perfect solution complete with diagrams. Kari writes:
The twist in this variation is that you are not actually trying to play three-in-a-row but trying to force the opposing player to play two-in-a-row with an empty third position in the row.
Nice insight. Here’s how player one wins.
Start in the center. The symbol doesn’t matter, so let’s say you begin with X. Now I cannot place an X in any free square because it would give you a three-in-a-row on the next turn. I must place an O and there are only two options: a corner cell or an edge cell (which corner or which edge doesn’t matter because the board is symmetric).
If I choose a corner cell, then you’ll place an O in the opposite corner:
Now I’m in zugzwang, which is a term meaning that every legal move puts me at a disadvantage. My best move would be to pass, but the rules obligate me to move. No matter what cell I pick and which symbol I place in it, I’ll open up a three-in-a-row for you.
If instead I place an O in an edge on my first move, then you’ll follow a similar strategy and place an O on the opposite edge:
Here, the only way I can avoid giving you a win on your next turn is to place an O in one of the remaining edge cells. Again you’ll mirror it and put me in zugzwang:
All moves for me lose. So the first player can always force a win in anarchy tic-tac-toe by opening in the center and then mirroring their opponents moves until they have a winning opportunity.
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