Note: Continued from an earlier post.
We decided that five was a pretty good starting number, and then we were starting to consider our opponent's best move.
1. What happens when you [the opponent] choose 1, 3, 7 or 9? Are they good / strong moves? Can I force a win after you chose one of those numbers?
Let me list all the 15-triplets again for reference in this post:
However, if I choose 6 or 8, in your next move you can also sabotage me easily by choosing 4 or 2.
All this get very confusing, and we have yet to find any clear winning strategy here. Everything is in a mess, and inevitably we start falling back to our "maths is convoluted" assumption again.
Okay, since the choice of 1, 3, 7 or 9 doesn't lead us anywhere, let's see what happens with 2, 3, 6 or 8.
2. What happens when you choose 2, 4, 6 or 8? Are they good / strong moves? Can I force a win after you chose one of those numbers?
If you choose any of these, you have three "good" triplets. For example, if you choose 2, your "good" triplets would be 2,4,9; 2,5,8 and 2,6,7. Again, 2,5,8 is no longer possible because I had taken 5. That leaves 2,4,9 and 2,6,7 as your "good" triplets.
With this, can I still force a win?
If I choose 1, you are forced to choose 9 in your next move (or else I will win). Then I am forced to choose 4. You have 3, 6, 7, 8 as your remaining "okay" moves. Then think of what will happen for each of these options.
I will spare you the details, but if you repeat the same analysis for all future steps (game-tree analysis), you would realize that all moves will end up with a draw, i. e. no one can get a 15-triplet.
3. In the end, can my initial choice of 5 force a win?
From our analysis, unfortunately, NO.
4. If the initial choice of 5 can't force a win, does that automatically mean that other initial choice can't force a win too?
All our analysis are based on the assumption that 5 is the best first number. However, we had no absolute proof that it MUST be the best first number. So with any other first numbers, it's still likely that some force-win strategies might be hidden in there somewhere.
But up to this stage, it's getting boring and repetitive. There must be some easier and better ways of solving this problem. But how lerr?
Will leave it for next time. If you are interested, please refer to Woon Pang's answer in my previous posts for ideas. :)