Prove Equal Probability Impossible with Relabeled Dice Faces
Problem Description
You are given two fair, six-sided dice. You have the ability to relabel the faces of each die with any integers, and duplicate labels are permitted on a single die. When a die is rolled, each of its six faces is equally likely to appear.
Task
Your task is to determine if it's possible to relabel the faces of these two dice such that when both dice are rolled, every possible sum from 1 through 12 occurs with precisely equal probability. If you believe it is possible, you must provide one valid labeling for both dice. If you conclude it is impossible, you must provide a rigorous mathematical proof of impossibility.
Constraints
- There are 36 total possible outcomes when rolling two dice (6 faces × 6 faces).
- For each sum from 1 through 12 to occur with equal probability, each of these 12 distinct sums must appear exactly 3 times out of the 36 total outcomes (36 / 12 = 3).
AI interview prep
Core dilemma
The central engineering conflict is proving the impossibility of achieving a perfectly uniform sum distribution (1-12, 3 times each) from two relabeled six-sided dice. This requires demonstrating that the inherent combinatorial properties of integer sums from a fixed number of outcomes fundamentally conflict with the desired uniform distribution.
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Interview prep
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Prove Equal Probability Impossible with Relabeled Dice Faces
Problem Description
You are given two fair, six-sided dice. You have the ability to relabel the faces of each die with any integers, and duplicate labels are permitted on a single die. When a die is rolled, each of its six faces is equally likely to appear.
Task
Your task is to determine if it's possible to relabel the faces of these two dice such that when both dice are rolled, every possible sum from 1 through 12 occurs with precisely equal probability. If you believe it is possible, you must provide one valid labeling for both dice. If you conclude it is impossible, you must provide a rigorous mathematical proof of impossibility.
Constraints
- There are 36 total possible outcomes when rolling two dice (6 faces × 6 faces).
- For each sum from 1 through 12 to occur with equal probability, each of these 12 distinct sums must appear exactly 3 times out of the 36 total outcomes (36 / 12 = 3).