Tag Archives: dice

Dice and Dissection: a puzzle

This is a puzzle, first appearing in Martin Gardner’s column in 1978, with a new way of thinking about the solution. Before the puzzle, though, a cultural diversion.

Rolling it in

In the modern classic board game The Settlers of Catan, it’s very important to know, when you roll a pair of dice, the frequency with which each number occurs. Resources are given players only if they have a settlement adjacent to those tiles whose number is rolled. If you build your settlements next to a tile labelled 2 or 12, it will, on average, only be productive once every 36 rolls. Tiles labelled 6 or 8 will produce resources five times in 36 rolls. It’s so fundamental to the gameplay that the relative frequencies are visualised as dots on the pieces: sixes and eights are so important, they are marked in red (rolling a seven does something different).

Puzzle

The well-known puzzle is:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

If I didn’t require you to use a positive number of spots on each face: then dice labelled {0, 1, 2, 3, 4, 5} and {2, 3, 4, 5, 6, 7} would work as a pair. If you allow negative numbers, there’s infinitely many solutions!

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All for one and none for all! Diversification, regulation and the tragedy of the commons.

Some proverbs come in contradictory pairs, for instance “Too many cooks spoil the broth” and “Many hands make light work”. I’d like to present an example that I feel illustrates two of these simultaneously. Everyone knows “Don’t put all your eggs in one basket”, including banks, which diversify by holding many different investments of different types. But while it may be in banks’ best interests to lower their levels of risk through diversification, it may plausibly raise the risk of a system-wide failure: if all banks follow that same advice, the rest of us may be “In for a penny, in for a pound”.

This is the argument given by Beale et al. in their paper Individual versus systemic risk and the Regulator’s Dilemma. Here’s a simple example that illustrates the main idea, adapted from a related comment in the earlier paper Systemic risk: the dynamics of model banking system.

Some people are playing a dice game. Each of them rolls a single fair die once and receives £1 for rolling a 1, £2 for a 2, and so on, up to £6 for a 6. However, afterwards they each have to pay £1.50 for the privilege of playing this rewarding game. Each of them starts out with no money, and so if they roll a 1, they go bankrupt.

If you play this game, there’s a \frac{1}{6} chance that you’d go bankrupt on your roll. Let’s say ten people play the game: the chance of all of them going bankrupt on their single throw is (1/6)^{10}, which is about one in sixty million.

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Intransitive votes

In the previous post I gave an example of students deciding who best guessed some lecturers’ ages.

I chose the numbers carefully so that under three reasonable methods of measuring:

Method  First place  Second place  Third place
Method a Adam Beth Charlie
Method b Beth Charlie Adam
Method c Charlie Adam Beth

This is actually almost identical to Condorcet’s voting paradox:

Voter  First preference  Second preference  Third preference
Voter 1  A  B  C
Voter 2  B  C  A
Voter 3  C  A  B

If three people in an election vote for candidates A, B, and C this way, then even using a method that takes account of all the preferences in one of the Condorcet voting system leads to a deadlock.  Continue reading

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