Roll three cube-shaped dice. What is the probability that the numbers on the top faces of the three dice form a triangle? For example, most people think that 1, 1, 2, if converted to lengths, will form a triangle, but they will not. Cut a plastic straw into 1 inch, 1 inch, and 2 inch lengths and see if they form a triangle! For the standard cube-shaped dice, the program computes the probability as 111/216. Set the slider to 4, 8, 12, and 20 to find the probabilities for each set of three dice.The probabilities may not be as high as intuition might predict.[There is a numerical relationship in the data. See Notes and Credits]
A computer program is used to collect data, the data is analyzed, and the method of 'finite differences' is used to describe the data using a Greatest Integer Function [GIF]. The result can be generalized to any set of three regularly-shaped dice with n faces. This problem has an interesting history. The year the Commodore 8K PET computer became available (1977) I happened to have lunch with Dr. Arthur Wiebe, the head of the Mathematics Project at Fresno Pacific University. He shared a probability problem he was having students work on. The problem asked what is the probability that the numbers on the top faces of three dice form a triangle? His students had tested, by hand, each of the 216 possible combinations (6 X 6 X 6). I shared that the dice used to play Dungeons and Dragons came in many different forms, some with fewer than six faces and some with more than six faces. Fpor example, an icosahedral die has twenty faces. Therefore, the total number of possible combinations for a set of three icosahedral dice is 20 X 20 X 20 = 8000. No one wants to work that out by hand! I told Dr. Wiebe that I would write a BASIC program on my Commodore PET to generate and test the three-number combinations for any set of regular dice with n faces! I gave the data to Dr. Wiebe, Dr. Wiebe gave it to Wil Reimer, a math instructor, and Wil gave the data to his students. The method of finite differences was applied to the data and a Greatest Integer Function was found to fit the data. The problem was extended to the probabilities for isosceles, equilateral, and scalene triangles. The computer program was modified, data collected, and again, the method of finite differences generated a GIF for each class of triangles. Problem generalized.
