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## 'Dicey Decisions' printed from http://nrich.maths.org/

This problem follows on
from Troublesome
Dice.
Jenny and her family often roll a die to decide who gets to eat the
last slice of pie, but when they roll it, the die often seems to
land "edge-up" or "corner-up" on a napkin, so they roll
again.

One day, Jenny suggests that instead of rolling again, when the die
lands corner-up or edge-up, they simply add the visible faces
together:

What assumptions would you need to make to come up with a fair way
of determining who gets the last slice of pie?

Here are some questions you might
like to consider:
Suppose the die lands face-up with probability $\frac {1}{2}$,
edge-up with probability $\frac{1}{3}$ and corner-up with
probability $\frac{1}{6}$. How could you allocate totals to three
people to give each of them an equal chance of
getting the last slice of pie?

Suppose the die lands face-up with probability $p$, edge-up with
probability $q$, and corner-up with probability $r = 1-(p+q)$. Are
there any family sizes for which you can always allocate totals
fairly, regardless of the values of $p$, $q$ and $r$?

The relative frequency of face-up, edge-up and corner-up
scores for the standard 6-sided die are distributed symmetrically.
Can you explain why?

This problem is based on an idea
shared by Paul Stephenson.