1. Understand the Problem
Goal: Find numberings for a 4-sided die and a 9-sided die so that when rolled, their sum distribution matches that of two standard 6-sided dice.
Standard 6-sided Dice:
Sum 2: 1 way
Sum 3: 2 ways
Sum 4: 3 ways
Sum 5: 4 ways
Sum 6: 5 ways
Sum 7: 6 ways
Sum 8: 5 ways
Sum 9: 4 ways
Sum 10: 3 ways
Sum 11: 2 ways
Sum 12: 11 way
2. Constraints
Positive Integers: Numbers on the dice must be positive whole numbers (1, 2, 3, ...).
Distinct or Non-Distinct: Numbers on each die can be repeated.
3. Approach
Systematic Trial and Error:
Start with a reasonable numbering for the 4-sided die (e.g., 1, 2, 3, 4).
Experiment with different numberings for the 9-sided die, adjusting to achieve the desired sum frequencies.
Use a spreadsheet or a similar tool to track the sum frequencies and make adjustments more efficiently.
4. Example Solution
4-sided die: 1, 2, 3, 4
9-sided die: 1, 1, 2, 3, 4, 5, 6, 7, 8
This combination can be shown to achieve the same sum distribution as two standard 6-sided dice.
