Can you find a set of cards among these six, such that the socks on the chosen cards can be grouped into matching pairs? (Duplicate pairs of the same sock are OK)
Spoilers
If the cards are indexed as
1 2 3
4 5 6
Then the following three subsets work: $\{ 1, 2, 4, 5, 6 \}$, $\{ 2, 3, 6 \}$, and $\{ 1, 3, 4, 5 \}$. There might be more but I don’t see them.
This is the objective of the game Socks, a card game originally designed by Anna Varvak, a math professor at Soka University of America. I tweaked the rules slightly, and designed the deck above, and you can buy a copy. (I’m selling it with Anna’s permission)
The game consists of 63 cards, one for each possible nonempty subset of 6 distinct socks. Duplicate pairs of the same kind of sock are OK, so long as they can be grouped into pairs (e.g., 3 yellow socks is not allowed, but 4 is). Players deal some cards and race to find sets, and the player at the end of the game with the most sets wins. The natural math question is: how many cards are needed to guarantee there’s a set?
Mathematically, the cards can hence be viewed as elements of the additive group $(\mathbb{Z}/2\mathbb{Z})^6$ of length-6 binary vectors. An index corresponds to a sock type, and the value is one of the sock is present on the card, and zero otherwise.
I use the “additive group” structure because a set in Socks is a subset of $(\mathbb{Z}/2\mathbb{Z})^6$ that sums to zero in the group. In programmer terms, a card is a length-6 bitstring and a “sum” is a bitwise XOR. Either way, “summing” two cards can be thought of as producing the card consisting of only the socks that show up an odd number of times on the two cards. One implication here is that any set requires at least three cards: each group element is its own inverse, but no card shows up twice in the deck.
Then the natural math question becomes: how many cards do you need to guarantee there is a valid set? In math terms, what is the smallest integer $M$ such that every size-$M$ subset $S \subset (\mathbb{Z}/2\mathbb{Z})^6$ contains a nonempty zero-summing subset?
Theorem: $M = 7$.
Proof. The following set of size 6 has no zero-summing subset
\[ \{ (1, 0, \dots, 0), (0, 1, 0, \dots, 0), \dots, (0, 0, \dots, 1) \} \]
So $M > 6$. On the other hand, consider any set $S$ of size 7. There are $2^{7} – 1$ distinct sums of nonempty subsets of $S$, but only $2^6$ group elements, so there must be two distinct subsets of $S$ that have the same sum. Suppose they are $X = {x_1, \dots, x_r}$ and $Y = {y_1, \dots, y_s}$, and $g = x_1 + \dots + x_r = y_1 + \dots + y_s$. Then adding the two equations, and noting that every element $x$ satisfies $x = -x$ in this group, we get
\[ x_1 + \dots + x_r + y_1 + \dots + y_s = g + g = 0 \]
In other words, we would like to use the “set” $\{ x_1, \dots, x_r, y_1, \dots, y_s \}$ and declare victory, but we can’t because some of the $x_i$ may coincide with some of the $y_j$. That is, $X$ and $Y$ can overlap, and we can’t “use” an element twice. But because the two sets are distinct (not equal), they cannot overlap completely. For any values that overlap, say $x_1 = y_2$, their sum is $x_1 + y_2 = 2x_1 = 0$, and those two elements can be removed without changing the sum. Hence, the final zero-summing subset is the symmetric difference $X \triangle Y$.
$\square$
Unfortunately, as I’ve played Socks, I’ve found that it’s no easier to find a zero-summing subset than it is to find two subsets that have the same sum. So while knowing this proof helps me win the hearts and minds of my opponents, it doesn’t help me win the game.
The next natural goal for a mathematician is to ask the same question of more general groups. The above proof argument naturally extends to $(\mathbb{Z}/2\mathbb{Z})^k$ having $M= k+1$. But for $(\mathbb{Z}/n\mathbb{Z})^k$, the problem is open.
Conjecture: Let $G = (\mathbb{Z}/n\mathbb{Z})^k$, then every set $S$ of $1 + k(n-1)$ elements of $G$ has a nonempty zero-summing subset.
The same argument from the previous theorem doesn’t quite apply: even though you can prove some distinct subsets of $S$ have the same sum, the difference gives different group elements that may not be in $S$.
This generalization opens the door to a decent subset of the number theory and additive combinatorics literature, in which this is called the Olson’s constant problem. Section F.3.3 of this survey of Bela Bajnok covers the literature quite well. Erdős originally worked on the problem in the 60’s. There have been a swath of results for groups with particular structure, for example in John Olson’s original 1969 paper, A Combinatorial Problem on Finite Abelian Groups, I, in which he proves the above conjecture for the special case of finite abelian $p$-groups. I will summarize a few results and conjectures below, most copied from Bajnok’s survey, and if you’re clever enough, each one could be the basis for a new card game. The hard part, it seems, is finding a theme that is cute and picking a small enough group so as to make it fun.
Theorem: For every even integer $n$, Olson’s constant for $\mathbb{Z}/n\mathbb{Z}$ is at least $1 + \lfloor \sqrt{2n – 3} \rfloor$.
For all even $n \leq 64$, this bound is known to be an equality. E.g., for $n=64$ it is 12, and for $n=50$ it is 10. There are no known values of $n$ for which this bound is not tight.
Theorem: For every prime $p$, Olson’s constant for $\mathbb{Z}/p\mathbb{Z}$ is exactly $1 + \lfloor \sqrt{2p} – 1 / 2 \rfloor$.
For example, for $p=53$, Olson’s constant is 10. (Deal ten cards containing…something both cute and interpretable mod 53! Good luck)
Odd composite values of $n$ is still an open problem, though a lower bound is known of $1 + \lfloor (\sqrt{8n + 9} – 1 ) /2 \rfloor$.
The case for products of cyclic groups involves various bounds (the $\tau$ below represents Olson’s constant minus 1).
Equality is known to hold for $k = 2, 3, 4, 5$.
Theorem: For any finite Abelian group of order $n$, Olson’s constant is less than $3 \sqrt{n} + 1$.
Conjecture [Erdős]: For any finite Abelian group of order $n$, Olson’s constant is less than $1 + \sqrt{2n}$.
Theorem: There is a constant $C$, such that for any finite Abelian group of order $n$, Olson’s constant is at most $1 + \sqrt{2n} + C \sqrt[3]{n} \log_e n$.
And finally, it is conjectured that cyclic groups have maximal Olson constant among all groups of a given order. So if you want to make your game hard, use cyclic groups. If you want to make it easy, use groups that are products of many small cyclic groups.
The origin story
The story of how I came to make my own version of Socks is a funny little mix-up.
Though the original game was designed by Anna Varvak in 2012, I made the cards shown in the picture at the beginning of this article. A friend of mine originally told me about Socks about six months ago. She had heard about it from a friend of hers, but she couldn’t find a physical copy for sale. Socks is listed at BoardGameGeek, but it had a broken website link, www.socksgame.com. I assumed it was out of print. I thought making my own version would be fun! So I found The Game Crafter, a website for on-demand printing of game components, and threw together a design.
When I finished, I noticed the generated URL was www.thegamecrafter.com/games/socks2 and I thought, “That’s weird, someone else made a game called Socks!” And voila, https://www.thegamecrafter.com/games/socks points to Varvak’s original game, still for sale.
Luckily, I was able to get in contact with Anna, and she graciously gave me permission to sell my version under the same name, “Socks.” The official rules in each game are slightly different: in hers you deal 12 cards and a set must consist of exactly 3 cards. In mine you deal 7 cards, and a set can be formed from any subset of the cards. When you restrict the subsets to have size exactly 3, the problem is slightly different (see section F.3.1 of Bajnok’s survey), and the main result that applies is
Theorem: For all positive integers $k$, every set $S \subset (\mathbb{Z}/2\mathbb{Z})^k$ with size at least $2^{k-1} + 2$ has a zero-summing subset of size 3, and there is a set of size $2^{k-1} + 1$ with no zero-summing subset of size 3.
In Anna’s version, it appears, 12 cards is not enough. You need a whopping 34 to guarantee a set exists.
Some playing notes
Some notes about what happens when actually playing the game:
- A set is worth 1 point, no matter how many cards are in it. I have no idea if it is a better strategy to look for smaller or larger sets.
- A set must have size at least three, and sets of size exactly three are easy-ish to find by XOR-summing pairs of cards.
- Each sock type is in the same location on the cards, which makes it easier to visualize.
- Often you can locate small groups of cards (say, 1-2) that are unusable because their inclusion forces an odd number of some sock. This seems to drastically diminish the size of the search space.
- A set using all the cards is not that uncommon, so it seems to help to start by summing all the socks and identifying the socks that break that possibility, or else claim a quick set.
- It does not appear to be easier to find sets when the cards are “sparse” (1-3 socks per card) vs “dense” (4-6 socks per card).
- When the last card is dealt you can immediately claim “Socks”, as long as every claimed set in the game was valid. This makes the last round boring, and it is slightly more interesting to leave the last card undealt, and then “infer” what is on that card by inverting the sum of the remaining cards. This is similar to the parlor trick for SET I wrote about a while back. Doing this trick for Socks is not nearly as hard as it is for SET, but might be impressive to players who have not thought about the math.