You’re given a delightful constraint puzzle:

You have two six-sided cubes. You can assign one digit per face, using digits 0–9.
Your task is to arrange digits on the cubes such that every day of the month from 01 to 31 can be displayed.
Importantly, for dates 01–09, a leading zero is required.

Digits can appear on both cubes, and the two cubes are placed side by side (in either order) to form each two-digit number.

Step 1: What Digits Are Needed?

We need to display every number from 01 to 31. So, we need to form:

  • 01 to 09 (leading zero)
  • 10 to 31

The digits required across all combinations:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9

So, both cubes must between them represent all 10 digits, possibly duplicating some.

Step 2: Logical Deduction

Let’s figure out what must go on the cubes by pure deduction:

  1. The Double Digits: To display 11 and 22, the digits 1 and 2 must be present on both cubes. (We don’t need 33 for a month calendar).
  2. The Zero: To display dates 01 through 09, a 0 must be paired with all 9 digits (1-9). Since a single cube only has 6 faces, it cannot hold all 9 digits to pair with a single 0. Therefore, the digit 0 must also be on both cubes to split the load.

At this point, we’ve used 3 faces on both cubes:

  • Cube 1: 0, 1, 2, _, _, _
  • Cube 2: 0, 1, 2, _, _, _

Step 3: The Missing Digits and The Trick

We have exactly 6 empty faces left across both cubes. The digits we still need to place are: 3, 4, 5, 6, 7, 8, 9 (7 digits).

This looks impossible since we need 7 digits but only have 6 faces!

This is where the classic puzzle trick comes in: allow the digit 6 to double as 9 by turning the cube upside down.

This reduces the remaining required digits to exactly 6: 3, 4, 5, 6, 7, 8.

We can distribute these 6 digits evenly across the 6 remaining empty faces. For example:

  • Place 3, 4, 5 on Cube 1
  • Place 6, 7, 8 on Cube 2

Final Answer

Cube 1: 0, 1, 2, 3, 4, 5
Cube 2: 0, 1, 2, 6, 7, 8
(Treat 6 as 9 when needed)

Reference