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I am reposting notzeb's solution to the 3 front case here, as I expect the 5 front case to be at least as badand making some comments on it. In particular, notice I will point out that the winning solution is given by a measure with not unique; while notzeb used fractal support.

Solution (rephrased): Take the triangle of possible solutions and inscribe a hexagon in itmethods, with vertices at the permutations of $(0,1/3,2/3)$. All our solutions I will be inside that hexagon.

Now, take that hexagon and place 6 smaller hexagons in it as shown belowgive a piecewise smooth solution using the same ideas.

Choose one

## Idea of those6hexagonsuniformlyatrandom.Place6smallerhexagonsinsidesolution:

I claim that one, and choose one of these uniformly at random again. Keep going. The hexagons shrink in size each time; the limiting point it is your army enough to find any probability distribution .