These games should always start at the end, to consider how you can ensure victory, and dividing the situations in which values \u200b\u200band values \u200b\u200bto win you lose.
In our case, we must begin the study from 100, which is the first time that someone wins. Obviously, 93 and 96 are positions from which we can win, we must prevent our opponent has them. Looking for a number with which to force our opponent to return that value, are suddenly 89. If our opponent adds 4, it outputs 93, and if you add 7, 96. In either of both, we win. So really, whoever gets to leave by 89 wins.
Reasoning backward, find the 78, 67, 56, 45, 34, 23, 12 and 1. If we can pass 1 to our rival, sure we would get to 100 and we would win.
Unfortunately, we must begin by adding 4 or 7. So we should not point to 100. If we repeat the system (note that is to subtract 11 each), we reason from 200 we should start from 2, to reach 300, from 3 to at last!, To get to 400, from 4.
Now let's check if our system works. We started by adding 4. Our opponent makes his move, and we do the opposite, so that they reach 15. We will continue, adding the move contrary to what our opponents do. When we get close to 100, we pass the rival 4 + 11 * 8 = 92, so that you can not get 100, if not 96 or 99. We add the number opposite, reaching 103, and proceed. The next step will be dangerous 103 + 8 * 11 = 191, in which our opponents can get to 195 or 198. Join us again to reach 202. The final step will be to dangerous 202 + 8 * 11 = 290, in which our opponents can go to 294 or 297. We arrive in any case at 301. The end of the game will arrive in the 301 + 8 * 11 = 389, which we pass to rival that number, so that brings us back 393 or 396, winning in any case to get 400.
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