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Two Dice = Nine Cards

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.

Previous Puzzle: Breaking a Chocolate

How many steps are required to break an m x n bar of chocolate into 1 x 1 pieces? We may break an existing piece of chocolate horizontally or vertically. Stacking of two or more pieces is not allowed.

Alice places three identical boxes on a table. She has concealed a precious ruby in one of them. The other two boxes are empty. Bob is allowed to pick one of the boxes. Among the two boxes remaining on the table, at least one is empty. Alice must then remove one empty box from the table. Finally, Bob is allowed to open either the box he picked, or the box lying on the table. If he opens the box with the ruby, he gets a kiss from Alice (which he values more than the ruby, of course). What should Bob do?

7 Oct 2012
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