# Strategy for a game of breaking sticks

Two persons have 2 uniform sticks with equal length which can be cut at any point. Each person will cut the stick into $n$ parts ($n$ is an odd number). And each person's $n$ parts will be permuted randomly, and be compared with the other person's sticks one by one. When one's stick is longer than the other person's, he will get one point. The person with more points will win the game. How to maximize the probability of winning the game for one of the person. What is the best strategy to cut the stick.

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How does someone win this game? For what reason are the stick fragments compared? – Arthur Fischer Aug 9 '12 at 12:38
@ArthurFischer Updated! – Fan Zhang Aug 9 '12 at 12:42
@Arthur: I suspect that the larger piece in each comparison scores a point, and the player with the larger number of points wins. – Brian M. Scott Aug 9 '12 at 12:42
@joriki Thank you, updated. – Fan Zhang Aug 9 '12 at 13:31

When intuition doesn't help, try brute force.

trials = Table[With[{k = RandomReal[{0, 1}, {7}]}, k/Total[k]], {50}]; Column[Sort[Transpose[{Total /@ Table[Total[Sign[trials[[a]] - trials[[b]]]], {a, 1, 50}, {b, 1, 50}], trials}]]]

Now we can look at some best/worst performers.

{-55, {0.018611, 0.0405574, 0.032157, 0.333219, 0.311017, 0.17885, 0.0855879}},
{-39, {0.313092, 0.178326, 0.0454452, 0.064321, 0.228231, 0.0907115,0.0798742}},

{29, {0.0360088, 0.220396, 0.13208, 0.145903, 0.180813, 0.240151, 0.044648}},
{29, {0.0799122, 0.0547817, 0.234127, 0.119589, 0.0290167, 0.255561,0.227013}},
{33, {0.0541814, 0.216338, 0.0619272, 0.204252, 0.0254828, 0.225743,0.212075}}

So, in the case of 7 pieces, best strategy seems to be to give 3 tiny pieces and 4 larger pieces. With some bigger trials, 2 tiny pieces and 5 larger pieces seemed best. But this is a competitive game ... so I selected the top 10% of a larger trial, and then had those guys compete. This time the winner was to have 3 values of 2/7 and one value of 1/7 {0.0263105, 0.0848388, 0.228165, 0.249932, 0.0788568, 0.215831, 0.116066}