Two Dice = Nine Cards
7 Oct 2012
Puzzle

Alice has two standard dice with labels 1 thru 6. When she rolls them and adds their labels, she gets a distribution over integers in [2, 12]. Bob has nine cards, each labeled with some real number. When Bob chooses two cards (without replacement) and adds their labels, he gets exactly the same distribution over integers in [2, 12] as Alice gets by rolling her dice. What are the labels on Bob's nine cards?

Source

Heard from a fellow Googler in 2012.

Solution

The numbers on the nine cards are { 0.5, 1.5, 2.5, 2.5, 3.5, 4.5, 4.5, 5.5, 6.5 }.

Followup

1. Charlie has two six-sided dice, which are standard in shape, but not standard in the number of spots on each face. Each face of Charlie's dice has a strictly positive number of spots, but neither is a standard die like each of Alice's. Charlie's dice however, also produce integers between 2 and 12 with exactly the same frequency of occurrence as Alice's method.
2. I have five pumpkins. I have the following weights for every pair of pumpkins: 10, 11, 12, 13, 14, 15, 16, 17, 18, 20. How much does each pumpkin weigh?
3. IBM Ponder This (October 2012):
There are seven polyhedral dice in a Dungeons & Dragons game. The result of throwing the 7 D&D dice is the sum of the seven numbers from each die:
• One tetrahedron: numbers 1, 2, 3, 4
• One cube: numbers 1, 2, 3, 4, 5, 6
• One octahedron: numbers 1, 2, 3, 4, 5, 6, 7, 8
• Two pentagonal trapezohedron: numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• One dodecahedron: numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
• One icosahedron: numbers 1, 2, 3, ..., 20.

Find a way to get the same distribution (same probability for each result) of tossing all seven D&D cubes by tossing four Disdyakis Triacontahedrons with integer numbers between 1 and 17 on their 120 faces. Please supply your answer as four lines of 17 numbers. Line i describes the i-th Disdyakis Triacontahedron and Column j describes how many times the number j appears on it.