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X-dice

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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