From Ed Carl comes a surprising game of dice:
We’re playing a game where you have to pick four whole numbers. Then I will roll four fair dice. If any two of the dice add up to any one of the numbers you picked, then you win! Otherwise, you lose.
For example, suppose you picked the numbers 2, 3, 4 and 12, and the four dice came up 1, 2, 4 and 5. Then you’d win, because two of the dice (1 and 2) add up to at least one of the numbers you picked (3).
To maximize your chances of winning, which four numbers should you pick? And what are your chances of winning?
There are $6^4 = 1296$ possibilities outcomes of 4 dice rolls.
There are $ {11 \choose 4} = 330$ ways of chosing 4 sums.
There will be a subset of outcomes remaining which is dependent on the sum chosen and highly variable.
I wrote some code. For each of the sum possibilities, it looked at all 1296 outcomes and successively eliminated outcomes won for the unique combination of 4 numbers.
Choosing the sums of 4, 6, 8, 10 yields $\dfrac{1264}{1296} \approx 97.53\%$
Choosing either 2, 6, 8, 10 or 4, 6, 8, 12 yields $1246$.
Note that 7, the sum that appears most often and in the highest number of rolls, is not optimal!
Choosing either 4, 6, 7, 9 or 5, 7, 8, 10 yields $1238$.