+27 When two standard 6-sided dice are rolled, there are 36 possible outcomes for the sum of the two rolls: one sum of 2, two sums of 3, and so on, up to one sum of 12.
Find all possible ways of numbering a 4-sided die and a 9-sided die with positive integers (not necessarily distinct), so that when they are rolled, the 36 possible outcomes for the sum of the two rolls are the same as the 36 possible outcomes for the sum of two standard 6-sided dice.
+1393 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.
