Marie, a puzzle enthusiast, has a unique set of dice that has piqued the interest of many. She possesses one regular die with the numbers 1, 2, 3, 4, 5, and 6, and a second cube that is completely blank. However, Marie has found a way to label this cube in a special manner that allows for intriguing gameplay.
When both dice are rolled, the possible total values that can be obtained are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. What makes this setup fascinating is that all 12 total values are equally likely to be rolled, adding an element of mystery and challenge to the game. The question that arises is – how did Marie label the cube to achieve this equilibrium in probabilities?
The solution to this puzzle lies in the labeling of the sides of the blank cube. The sides of the cube must be carefully labeled as 0, 0, 0, 6, 6, and 6. By assigning these values to the sides of the cube, Marie ensures that when both dice are rolled, the total sum outcomes are evenly distributed across all possibilities.
To understand the logic behind this labeling, let’s delve deeper into the mathematics involved. With two dice being rolled, there are a total of 36 possible combinations that can be obtained. For the 12 sum values to have an equal probability of occurring, each value must appear three times in the outcomes.
This unique labeling of the cube allows for the value of 12 to be achieved only when both dice show a 6. Since the value 12 must occur three times in the outcomes, it is imperative that the cube has three sides labeled with a 6. The remaining sides are labeled with 0 to ensure the required distribution of probabilities.
For each sum value from 1 to 12, the probability (p) is calculated as 3/36, which simplifies to 1/12 or approximately 8.3%. This uniform distribution of probabilities adds an element of unpredictability and excitement to the game, making it a challenging and engaging experience for players.
If you’re intrigued by this puzzle or have similar mind-bending challenges to share, feel free to reach out to us at games@sciam.com. This puzzle originally appeared in Spektrum der Wissenschaft and has been reimagined with permission to stimulate your problem-solving skills and mathematical reasoning.
