设置（游戏）

Set是一款游戏的游戏，每个卡都包含四个属性，每个属性都采用三个可能的值之一。游戏的目标是找到sets（因此，游戏的名称）三张牌，因此对于四个属性中的每一个，所有三张卡都有不同的值，或者所有三张卡都具有相同的值。

The cards have different numbers, colors, shapes, and textures, forming a set

Each card contains four attributes, each of which take on three values:

• Number: each card contains either 1, 2, or 3 shapes.
• Color：每张卡上的形状是红色，绿色或紫色。
• 形状t:he shapes on each card are either ovals, diamonds, or squiggles.
• Texture：每个形状都是空心，阴影或填充的。

Each combination of attributes corresponds to exactly one card, for a total of 81 cards in the deck. At any time, 12 cards are revealed, and the fastest player to find a set within these 12 cards wins the set (depending on the players, the penalty for a false set claim can range from being unable to claim a set again that round to losing an already won set). The three cards in the set are then removed, and another three cards are revealed, with the game continuing until the deck is completed.

Additionally, if no player is able to find a set within the group of 12 cards after some time, another three cards are revealed (with the agreement of all players). The next set claimed will not be replaced, since there will already be at least 12 cards showing. As shown below, it is indeed possible (though somewhat unlikely) that a group of 12 cards contains no set; in fact, it is possible for a group of up to20卡不包含集合。

尽管集合游戏高度基于模式识别，但有许多用于加快搜索过程的方法。

分析董事会：

Firstly, chances are very high that some attribute will be under-represented, in the sense that it is quite likely that (for example) there will be only one or two shaded cards in play. This means that sets involving those cards can be checked quite quickly, and if (as is most likely) they are not involved in any sets, any three cards that form a set necessarily have the same value for that attribute: in the previous example, any set would have three cards with filled shapes or three cards with hollow shapes.

同样，从某种意义上说，（例如）很有可能有很多带有椭圆形的卡片，可能会有很多属性会过分代表。这意味着通常值得仅检查包含这些卡的可能集，因为这可以确保一个属性得到满足（并且还会有更多可能满足其余三个属性）。

Transitioning between boards:

游戏中最重要的部分是透露三张新卡的时期，因为玩家已经对已经显示的9张卡进行了大量分析。

Firstly, it is not unusual for the remaining 9 cards to themselves contain a set, so it is certainly worth analyzing the remaining cards for this reason alone. However, even more can be learned by anticipating possible useful cards: for instance, an over-represented attribute (e.g. a lot of ones) within the 9 cards means that an additional card with the same attribute is very likely to complete a set.

最后，当揭示新卡时，它们可能是该集合所涉及的（如果存在），因为玩家将已经分析了9张显示卡片并从此卡片中丢弃（或删除）集合。

在没有设置的情况下，添加三张卡的机制导致了两个自然问题：

• 添加三张卡的可能性是多少？
• 可以揭示的最大数量的卡片是什么，使它们之间没有设置？

这两个问题都是通过用几何解释来解释集的游戏来回答的：每张卡都可以看作是 $\mathbb{F}_3^4$，这基本上意味着形式的4D点 $(a_1, a_2, a_3, a_4)$where $a_i=1,2,$或3 $i$. For instance, the point $1, 2, 3, 1$可能对应于“一个绿色空心skiggle”。

The geometric interpretation of a set is then quite nice: three cards form a set if their associated points form a line.

解决4个属性的情况仍然是必要的，但是当只有2个属性时，情况很容易分析（使空间置于空间 $\mathbb{F}_3^2$）。在such a situation, the question of the largest number of cards that contains no set is answered by the 2-dimensional geometric interpretation: what is the maximum number of integer points $x,y$with $0 \leq x \leq 2, 0 \leq y \leq 2$such that no three form a "line", where a line can "loop around"?

All three colors represent lines in $\mathbb{F}^2$

It is not too difficult to show that the maximal number of points (squares in the above picture) that do not form any line is 4:

Two examples of four points with no lines

Suppose it were possible to find 5 points, no three of which are collinear. Then each horizontal line contains at most 2 points, and so some horizontal line contains exactly one point (call it $P$）。

There are four lines going through $P$:

• $H$, the horizontal line through the point
• $V$，通过该点的垂直线
• $D_-$, the down-right diagonal line through the point
• $D_+$, the up-right diagonal line through the point

自从 $H$除了没有任何点 $P$, and each point other than $P$正是这些行之一，正是 $v，d_+，d_-$contains two points other than $P$(by the鸽子原理），因此包含三个共线点 - 矛盾。

Hence there is no collection of 5 points in $\mathbb{F}_3^2$with no three collinear, making the maximum 4.

这种推理在更高的维度中更难使用，但总体策略仍然相同。在 $n$dimensions, the maximum number of cards with no set is:

which, as applied to the traditional set game, means that it is possible to find 20 cards with no set^[1].

20 cards that contain no set

Even higher dimensions are more difficult to work with and require additional tools from the field ofRamsey theory. A recent (as of 2016) breakthrough shows that the maximal size of a cap set (a collection of cards with no set) in $n$尺寸最多有大小 $\ left（\ frac {2.756} {3} \ right）^n$，其应用程序更为迅速矩阵乘法algorithms.

The other question, regarding the probability of this occurring, is best answered by computer program. Knuth provides the following numbers^[2]。：

在典型的12卡情况下，它的答案略低于 $\ frac {1} {30}$.

1. Davis, B., & Maclagan, D.纸牌游戏集. Retrieved April 2nd, 2016, fromhttps://web.archive.org/web/20130605073741/http://www.math.rutgers.edu/~maclagan/papers/set.pdf
2. joriki, ?.在纸牌游戏集中，n张卡中存在的集合的可能性是多少？. Retrieved April 2nd, 2016, fromhttp://math.stackexchange.com/q/203146