Trying to break the 7 on 9 game

    Today there will be a small sketch just to understand whether you are interested in this. In general, there is such an uncle Aleksey Savvateev who reads an amazing course on game theory. He also wrote the book “Mathematics for the Humanities”, where he very subtly went over all the humanities. It’s just that mathematics, where the numbers from the formulas are rapidly disappearing. And then, one fine moment, I dragged him a 7 on 9 game and asked him to break it.

    The principle of the game is simple: cards are thrown into the middle of the table. Each card has a number and a difference. For example, 6 plus or minus 2 - you can throw 8 or 4 on top. The task is to throw all your cards in the center as quickly as possible. There is no order of moves, there is nothing, just who managed - he threw the card. Cards on one of his deck are taken out on hand, you can take any amount. At least that's it. The main thing is to have enough fingers.

    Here's what happened, video (below). Now I will tell you in words what it was (you can not read if you watched the video).

    First hypothesis

    You can sort the cards by color. The color determines the difference: the green ones give plus or minus 1, the blue ones give plus or minus 2, the red ones give plus or minus 3. Next, the task is simply to find the right moment, throw a green card there, throw all the other green cards quickly and quickly (I remind you, they differ by 1, that is, it’s very easy to sort), and then how it goes.

    Why is it good - because the opponents do not have time to read the card and think correctly while you throw them away. Any ligaments from the hand are played almost continuously and give the advantage that the opponent (or opponents) carefully stick all this time. More precisely, they overestimate the situation and consider it anew every time.

    Second hypothesis

    It is assumed that you can collect on your hand from all cards a buffer that will be lowered to the table at the right time. That is, to build a row precisely in the hands, and then lay out cards one by one, then you will win with one masterful action.

    Third hypothesis

    We need to play as it turns out, just laying out the cards according to the situation.

    In the video, we tried all three methods in a three-player game and two methods (second and third) in a two-player game. The game of reaction wins (the same was confirmed by about 20 more experiments) in some cases.

    Playing the buffer gives an undeniable advantage in the following situations:

    1. If you play strictly together, then you, in fact, both collect this same buffer of cards coming in a row from hand. Only the one who plays the reaction collects this with the fastest greedy algorithm, and you collect with something more complex, which allows you to get a greater chance of using almost all the cards with your hands.
    2. When the opponent pushes himself into a situation that he cannot put more cards (usually this happens with 3-5 cards on his hand), you need to spend 10 minutes drawing up his winning combination and lay it out.

    Realistic scenario? Quite. The game, of course, gets killed to hell as a process, but here we break it, and not have fun.

    What is the problem? In the fact that, as we see in the example of the second game, Max (the opponent of Alexei, who uses the buffer caching tactics) realized what would happen now, and calculated the optimal interception situations for himself. And he managed to insert a card between the actions of Alexei, which broke his entire buffer.

    Buffer interception can only be done in such a way as to knock down the buffer. That is, it makes little sense to intercept the green (Δ1) card with the blue (Δ2). Assume in the buffer 4 ± 1, then 5 ± 2. When intercepting 4 ± 1 with a card of 3 ± 2, the opponent simply continues his actions.

    What happens when playing three together? Just two generate enough matches to win or stay with one or two cards, which almost always means a true interception with a victory.

    In a box of 73 cards with numbers, the distribution of 1-10 is uneven. One card lays on the table. When playing for two, we get half of 72 cards. That is, the probability of flattening the buffer is not 100%. Since it’s ideal for us not just a buffer from a random card (more precisely, two random ones - the first and the last, because the buffer is played in both directions), but a cyclic buffer that guarantees victory, in practice the chances are even less. In fact, then the game in our scenario turns into a competition of the greedy algorithm (the second player collects the longest chain on the table) and something like annealing or branches and borders (in our hands). As a purely research task this is interesting, but the practical result is that it’s better not to break the buzz with others and quickly count while playing this game.

    All. If suddenly you want this box, then know that Alex tried to put his books on our network, but could not (because the contract of delivery with an individual is an unreal hell), and just gave us 32 pieces. We will be happy to give them in the store on Taganskaya (one in each hand) to those who say, “I am from our mutual friend Savvateev.”

    Game theory course here . UPD: lexnekr advises another course " Mathematics for all ."

    I separately recommend a short video about a duel of three persons:

    If you are interested in applying mathematics to games, we will continue to try to break something more complex. And yet this is not always about mathematics, for example, in yesterday’s “crashing” tests, we tried to take a new game with an unexpected agreement between the players where it would seem impossible to agree. Although this is also part of the theory of games and applications.

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