Designing Calendar Dice for Month Days

This puzzle asks you to design two nonstandard dice so that, when placed side by side, they can show any calendar day of the month as a two-digit number. You must think about which digits are truly necessary, how often they appear across the possible days, and when digits must be present on both dice rather than just one. The twist that single-digit days are still written with a leading zero means you always need to use both dice, and you can rely on being able to swap their positions. The heart of the puzzle is to find a compact, efficient layout of digits on the faces that still covers all day numbers.

Solving it leans heavily on combinatorial design and constrained optimization ideas. It rewards recognizing symmetry and redundancy among required two-digit combinations, and spotting opportunities where one symbol can play multiple roles depending on orientation. Interviewers watch for systematic case analysis rather than trial-and-error guessing, the ability to argue that some digits must be shared across both dice, and a clean justification that no required date has been missed and no die face is wasted.

What it tests

This problem class is governed by the principle of minimal representation under combinatorial constraints. When you must represent all two-digit combinations from a limited set of digits, and each digit must be assigned to a limited number of slots (here, die faces), the challenge is to maximize coverage by exploiting symmetries, overlaps, and dual-purpose assignments. The key is to identify digits that must appear on both objects due to repeated double-digit days (like 11 or 22), and to recognize when digits can serve dual roles (such as 6 doubling as 9 when orientation allows). The structure is about partitioning the required pairs into feasible assignments, using the fewest resources possible, while ensuring every required combination is constructible. This is a classic instance of combinatorial design, where the constraints force creative reuse and careful allocation.