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A new type of 6-sided die,
called an X-die, is proposed where instead of the faces being
numbered 1 to 6 as usual, the faces are numbered with positive
whole numbers such that their sum is 21. In this problem we will
say that a die $A$ is worse than a die $B$ if and only
if $P(A< B) > P(B< A)$ for a single throw. Conversely, a
die $A$ is better than a
die $B$ if and only if $P(A< B) < P(B< A)$ for a single
throw.Can you create an X-die which is worse than an ordinary die?
Can you create an X-die which is better than an ordinary die
using only the numbers $1$ to $6$ (you don't have to use all of the
numbers!)?
Be clear in your explanations or reasoning.
You can prove your results using algebra or explore the
problem experimentally using a spreadsheet.
Extension: Explore the
notion of a 'worst' or a 'best' X-die.