I am trying to understand the logic of the community in choosing a specfic strategy to maximize profit in a game. The game can be described as follows:
Every ticket has nine spaces, with a number which is uniformly randomly distributed among the 9 spaces on the board without duplication. At the start, however, eight of these nine spaces will be hidden [one number is given at a random space on the board]. To begin, select three numbers from the eight hidden on your ticket. Next, select one of eight lines─vertical, horizontal, or diagonal. When selecting a line, the sum of the three numbers in that line will determine the amount of [currency] you receive. After you have selected a line, all the numbers are uncovered and you will receive [currency] based on the sum of the line you chose.
The sums and respective payouts are as follows:
Sum Payout
--------------
06 10000
07 00036
08 00720
09 00360
10 00080
11 00252
12 00108
13 00072
14 00054
15 00180
16 00072
17 00180
18 00119
19 00036
20 00306
21 01080
22 00114
23 01800
24 03600
As a visual, the begining board may look like this, where x's represent spaces whose nunbers are hidden and the arrows represent the columns/rows/diagonal you can pick to sum.
↘ V V V ↙
> x x x
> x x x
> x 6 x
Likewise, chosing to uncover the board chosing the "Y" and "L" strategies respectively are as follows:
1 x 8
x 5 x
x 6 x
1 x x
4 x x
9 6 x
The logic behind the "Y" strategy is to determine wheather or not the best payout (see the 6 sets below) is possible. I do not know much about the logic of the "L" strategy more than is just being one which is preferred.
I have watched some videos about decision theory, which may help determine if one of the two strategies is "better", if that is possible. Specifically, videos regarding probability, expected payout, and "perfect information" should apply here.
1) Is there an easy way to find out the probability of each sum without having to do it by hand/program? The sums which are less likely have a higher payout than sums that are more likely to occur. There is only 6 ways to get a sum of 6 - by suming {1,2,3}, {1,3,2}, {2,3,1}, {2,1,3}, {3,2,1}, or {3,1,2}.
2) If the board shown by the "Y" and "L" patterns are the same (unknown to the player), how do the values of expected value for perfect information of each space compare to each other? Does the "Y" pattern give more information than the "L" pattern in terms of which numbers should be summed?
3) If the known space is in a corner, does a "Y" pattern rotated or mirrored give equivalent information as a straight "Y"? What about the "L" pattern?
To actually try to "solve" which strategy is better would probably take an algorithm. Currently, I'm just trying to understand how to approach the problem.