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# X-dice

##### Age 16 to 18 Challenge Level: 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