# Socks, a matching game based on an additive combinatorics problem

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.

# A Rook Game

Problem: Two players take turns moving a rook on an 8×8 chessboard. The rook is only allowed to move south or west (but not both in a single turn), and may move any number of squares in the chosen direction on a turn. The loser is the player who first cannot move the rook. What is the optimal play for any starting position?

Solution: Take advantage of the symmetry of the board. If the rook is not on the diagonal, the optimal strategy is to move it to the diagonal. Then when the other player moves it off, your next move is to move it back to the diagonal. If your opponent starts their turn with the rook always on the diagonal, then you will never lose, and by the symmetry of the board you can always move the rook back to the diagonal. This provides an optimal algorithm for either player. In particular, if the rook starts on a square that is not on the diagonal, then player 1 can guarantee a win, and otherwise player 2 can.

Symmetry is one of the most powerful tools in all of mathematics, and this is a simple albeit illustrative example of its usage.

# Want to make a great puzzle game? Get inspired by theoretical computer science.

Two years ago, Erik Demaine and three other researchers published a fun paper to the arXiv proving that most incarnations of classic nintendo games are NP-hard. This includes almost every Super Mario Brothers, Donkey Kong, and Pokemon title. Back then I wrote a blog post summarizing the technical aspects of their work, and even gave a talk on it to a room full of curious undergraduate math majors.

But while bad tech-writers tend to interpret NP-hard as “really really hard,” the truth is more complicated. It’s really a statement about computational complexity, which has a precise mathematical formulation. Sparing the reader any technical details, here’s what NP-hard implies for practical purposes:

You should abandon hope of designing an algorithm that can solve any instance of your NP-hard problem, but many NP-hard problems have efficient practical “good-enough” solutions.

The very definition of NP-hard means that NP-hard problems need only be hard in the worst case. For illustration, the fact that Pokemon is NP-hard boils down to whether you can navigate a vastly complicated maze of trainers, some of whom are guaranteed to defeat you. It has little to do with the difficulty of the game Pokemon itself, and everything to do with whether you can stretch some subset of the game’s rules to create a really bad worst-case scenario.

So NP-hardness has very little to do with human playability, and it turns out that in practice there are plenty of good algorithms for winning at Super Mario Brothers. They work really well at beating levels designed for humans to play, but we are highly confident that they would fail to win in the worst-case levels we can cook up. Why don’t we know it for a fact? Well that’s the $P \ne NP$ conjecture.

Since Demaine’s paper (and for a while before it) a lot of popular games have been inspected under the computational complexity lens. Recently, Candy Crush Saga was proven to be NP-hard, but the list doesn’t stop with bad mobile apps. This paper of Viglietta shows that Pac-man, Tron, Doom, Starcraft, and many other famous games all contain NP-hard rule-sets. Games like Tetris are even known to have strong hardness-of-approximation bounds. Many board games have also been studied under this lens, when you generalize them to an $n \times n$ sized board. Chess and checkers are both what’s called EXP-complete. A simplified version of Go fits into a category called PSPACE-complete, but with the general ruleset it’s believed to be EXP-complete [1]. Here’s a list of some more classic games and their complexity status.

So we have this weird contrast: lots of NP-hard (and worse!) games have efficient algorithms that play them very well (checkers is “solved,” for example), but in the worst case we believe there is no efficient algorithm that will play these games perfectly. We could ask, “We can still write algorithms to play these games well, so what’s the point of studying their computational complexity?”

I agree with the implication behind the question: it really is just pointless fun. The mathematics involved is the very kind of nuanced manipulations that hackers enjoy: using the rules of a game to craft bizarre gadgets which, if the player is to surpass them, they must implicitly solve some mathematical problem which is already known to be hard.

But we could also turn the question right back around. Since all of these great games have really hard computational hardness properties, could we use theoretical computer science, and to a broader extent mathematics, to design great games? I claim the answer is yes.

[1] EXP is the class of problems solvable in exponential time (where the exponent is the size of the problem instance, say $n$ for a game played on an $n \times n$ board), so we’re saying that a perfect Chess or Checkers solver could be used to solve any problem that can be solved in exponential time. PSPACE is strictly smaller (we think; this is open): it’s the class of all problems solvable if you are allowed as much time as you want, but only a polynomial amount of space to write down your computations.

## A Case Study: Greedy Spiders

Greedy spiders is a game designed by the game design company Blyts. In it, you’re tasked with protecting a set of helplessly trapped flies from the jaws of a hungry spider.

A screenshot from Greedy Spiders. Click to enlarge.

In the game the spider always moves in discrete amounts (between the intersections of the strands of spiderweb) toward the closest fly. The main tool you have at your disposal is the ability to destroy a strand of the web, thus prohibiting the spider from using it. The game proceeds in rounds: you cut one strand, the spider picks a move, you cut another, the spider moves, and so on until the flies are no longer reachable or the spider devours a victim.

Aside from being totally fun, this game is obviously mathematical. For the reader who is familiar with graph theory, there’s a nice formalization of this problem.

The Greedy Spiders Problem: You are given a graph $G_0 = (V, E_0)$ and two sets $S_0, F \subset V$ denoting the locations of the spiders and flies, respectively. There is a fixed algorithm $A$ that the spiders use to move. An instance of the game proceeds in rounds, and at the beginning of each round we call the current graph $G_i = (V, E_i)$ and the current location of the spiders $S_i$. Each round has two steps:

1. You pick an edge $e \in E_i$ to delete, forming the new graph $G_{i+1} = (V, E_i)$.
2. The spiders jointly compute their next move according to $A$, and each spider moves to an adjacent vertex. Thus $S_i$ becomes $S_{i+1}$.

Your task is to decide whether there is a sequence of edge deletions which keeps $S_t$ and $F$ disjoint for all $t \geq 0$. In other words, we want to find a sequence of edge deletions that disconnects the part of the graph containing the spiders from the part of the graph containing the flies.

This is a slightly generalized version of Greedy Spiders proper, but there are some interesting things to note. Perhaps the most obvious question is about the algorithm $A$. Depending on your tastes you could make it adversarial, devising the smartest possible move at every step of the way. This is just as hard as asking if there is any algorithm that the spiders can use to win. To make it easier, $A$ could be an algorithm represented by a small circuit to which the player has access, or, as it truly is in the Greedy Spiders game, it could be the greedy algorithm (and the player can exploit this).

Though I haven’t heard of the Greedy Spiders problem in the literature by any other name, it seems quite likely that it would arise naturally. One can imagine the spiders as enemies traversing a network (a city, or a virus in a computer network), and your job is to hinder their movement toward high-value targets. Perhaps people in the defense industry could use a reasonable approximation algorithm for this problem. I have little doubt that this game is NP-hard [2], but the purpose of this article is not to prove new complexity results. The point is that this natural theoretical problem is a really fun game to play! And the game designer’s job is to do what game designers love to do: add features and design levels that are fun to play.

Indeed the Greedy Spiders folks did just that: their game features some 70-odd levels, many with multiple spiders and additional tools for the player. Some examples of new tools are: the ability to delete a vertex of the graph and the ability to place a ‘decoy-fly’ which is (to the greedy-algorithm-following spiders) indistinguishable from a real fly. They player is usually given only one or two uses of these tools per level, but one can imagine that the puzzles become a lot richer.

[2]: In the adversarial case it smells like it’s PSPACE-complete, being very close to known PSPACE-hard problems like Cops and Robbers and Generalized Geography

## Examples

I can point to a number of interesting problems that I can imagine turning into successful games, and I will in a moment, but before I want to make it clear that I don’t propose game developers study theoretical computer science just to turn our problems into games verbatim. No, I imagine that the wealth of problems in computer science can serve as inspiration, as a spring board into a world of interesting gameplay mechanics and puzzles. The bonus for game designers is that adding features usually makes problems harder and more interesting, and you don’t need to know anything about proofs or the details of the reductions to understand the problems themselves (you just need familiarity with the basic objects of consideration, sets, graphs, etc).

For a tangential motivation, I imagine that students would be much more willing to do math problems if they were based on ideas coming from really fun games. Indeed, people have even turned the stunningly boring chore of drawing an accurate graph of a function into a game that kids seem to enjoy. I could further imagine a game that teaches programming by first having a student play a game (based on a hard computational problem) and then write simple programs that seek to do well. Continuing with the spiders example they could play as the defender, and then switch to the role of the spider by writing the algorithm the spiders follow.

But enough rambling! Here is a short list of theoretical computer science problems for which I see game potential. None of them have, to my knowledge, been turned into games, but the common features among them all are the huge potential for creative extensions and interesting level design.

### Graph Coloring

Graph coloring is one of the oldest NP-complete problems known. Given a graph $G$ and a set of colors $\{ 1, 2, \dots, k \}$, one seeks to choose colors for the vertices of $G$ so that no edge connects two vertices of the same color.

Now coloring a given graph would be a lame game, so let’s spice it up. Instead of one player trying to color a graph, have two players. They’re given a $k$-colorable graph (say, $k$ is 3), and they take turns coloring the vertices. The first player’s goal is to arrive at a correct coloring, while the second player tries to force the first player to violate the coloring condition (that no adjacent vertices are the same color). No player is allowed to break the coloring if they have an option. Now change the colors to jewels or vegetables or something, and you have yourself an award-winning game! (Or maybe: Epic Cartographer Battles of History)

An additional modification: give the two players a graph that can’t be colored with $k$ colors, and the first player to color a monochromatic edge is the loser. Add additional move types (contracting edges or deleting vertices, etc) to taste.

### Art Gallery Problem

Given a layout of a museum, the art gallery problem is the problem of choosing the minimal number of cameras so as to cover the whole museum.

This is a classic problem in computational geometry, and is well-known to be NP-hard. In some variants (like the one pictured above) the cameras are restricted to being placed at corners. Again, this is the kind of game that would be fun with multiple players. Rather than have perfect 360-degree cameras, you could have an angular slice of vision per camera. Then one player chooses where to place the cameras (getting exponentially more points for using fewer cameras), and the opponent must traverse from one part of the museum to the other avoiding the cameras. Make the thief a chubby pig stealing eggs from birds and you have yourself a franchise.

For more spice, allow the thief some special tactics like breaking through walls and the ability to disable a single camera.

This idea has of course been the basis of many single-player stealth games (where the guards/cameras are fixed by the level designer), but I haven’t seen it done as a multiplayer game. This also brings to mind variants like the recent Nothing to Hide, which counterintuitively pits you as both the camera placer and the hero: you have to place cameras in such a way that you’re always in vision as you move about to solve puzzles. Needless to say, this fruit still has plenty of juice for the squeezing.

### Pancake Sorting

Pancake sorting is the problem of sorting a list of integers into ascending order by using only the operation of a “pancake flip.”

Just like it sounds, a pancake flip involves choosing an index in the list and flipping the prefix of the list (or suffix, depending on your orientation) like a spatula flips a stack of pancakes. Now I think sorting integers is boring (and it’s not NP-hard!), but when you forget about numbers and that one special configuration (ascending sorted order), things get more interesting. Instead, have the pancakes be letters and have the goal be to use pancake flips to arrive at a real English word. That is, you don’t know the goal word ahead of time, so it’s the anagram problem plus finding an efficient pancake flip to get there. Have a player’s score be based on the number of flips before a word is found, and make it timed to add extra pressure, and you have yourself a classic!

The level design then becomes finding good word scrambles with multiple reasonable paths one could follow to get valid words. My mother would probably play this game!

### Bin Packing

Young Mikio is making sushi for his family! He’s got a table full of ingredients of various sizes, but there is a limit to how much he can fit into each roll. His family members have different tastes, and so his goal is to make everyone as happy as possible with his culinary skills and the options available to him.

Another name for this problem is bin packing. There are a collection of indivisible objects of various sizes and values, and a set of bins to pack them in. Your goal is to find the packing that doesn’t exceed the maximum in any bin and maximizes the total value of the packed goods.

I thought of sushi because I recently played a ridiculously cute game about sushi (thanks to my awesome friend Yen over at Baking And Math), but I can imagine other themes that suggest natural modifications of the problem. The objects being packed could be two-dimensional, there could be bonuses for satisfying certain family members (or penalties for not doing so!), or there could be a super knife that is able to divide one object in half.

I could continue this list for quite a while, but perhaps I should keep my best ideas to myself in case any game companies want to hire me as a consultant. 🙂

Do you know of games that are based on any of these ideas? Do you have ideas for features or variations of the game ideas listed above? Do you have other ideas for how to turn computational problems into games? I’d love to hear about it in the comments.

Until next time!

# Classic Nintendo Games are NP-Hard

The heroes and heroines of classic Nintendo games.

Problem: Prove that generalized versions of Mario Brothers, Metroid, Donkey Kong, Pokemon, and Legend of Zelda are NP-hard.

Solutionhttp://arxiv.org/pdf/1203.1895v1.pdf

Discussion: Three researchers (including Erik Demaine, a computer science professor at MIT famous for his work with the mathematics of origami) recently finished a paper giving the complexity of a number of classic Nintendo games (the ones I loved to play). All are proven NP-hard, some are shown to be NP-complete, and some are PSPACE-complete. Recall, a problem is NP-hard if an NP-complete problem reduces to it, and a problem is NP-complete if it’s NP-hard and also in NP. As we have just posted a primer on NP-completeness and reduction proofs, this paper is a fun next step for anyone looking for a more detailed reduction proof. A pre-print is available for free on arXiv, and it’s relatively short and easy to read. I’ll summarize his result here, and leave most of the details to the reader. Each game is “generalized” to the task of determining whether one can get from a “start” location to a “finish” location. So the decision problem becomes: given a finite level of a game, can the player move from the starting location to the finishing location? All of the reduction proofs are from 3-Sat, and they all rely on a common framework which can be applied to any platform game. Pictorially, the framework looks like this:

The framework for reducing 3-Sat to platform games.

The player starts in the “start” gadget, which allows one to set up initial state requirements. For instance, in Super Mario Brothers, the start provides you with a mushroom, and you cannot get to the finish without being able to break blocks by jumping under them, which requires the mushroom power-up. Each “variable” gadget requires the player to make a variable assignment in such a way that the player can never return to that gadget to make a different decision. Then each “clause” gadget can be “unlocked” in some way, and each clause gadget can only be visited by the player once the player has chosen a satisfying variable assignment for that clause. Once the player has visited all variable gadgets, he goes to the “check in” area, and can travel back through all of the clauses to the finish if and only if he unlocked every clause. The crossover of the paths in the picture above requires another gadget to ensure that the player cannot switch paths (the details of this are in the paper).

For example, here is the variable gadget described in the paper for The Legend of Zelda, a Link to the Past:

The variable gadget for A Link to the Past.

Note that here we require Link has the hookshot, which can grapple onto chests, but has limited reach. The configuration of the chests requires him to choose a path down one of the two columns at the bottom, and from there he may never return.

Here’s another example. In the classic Super Mario Brothers game, a possible clause gadget is as follows.

The clause gadget for the original Super Mario Brothers.

Note that if the player can only enter through one of the three columns at the top, then the only thing he can do is kick a red koopa shell down so that it breaks the blocks, unlocking the way for Mario to pass underneath at the end. Note that Mario cannot win if he falls from the top ledge (since must always remain large, he can’t fit through a one-tile-high entryway). Further details include the hole at the bottom, in which any stray koopa shell will necessarily fall, but which Mario can easily jump over. We recommend reading the entire paper, because it goes into all of the necessary details of the construction of the gadgets for all of the games.

## Future Work

We note that there are some parts of the paper that only got partial results, mostly due to the variation in the game play between the different titles. For instance, the original Super Mario Brothers is known to be NP-complete, but the added ability to pick up koopa shells in later Super Mario Brothers games potentially makes the decision problem more complex, and so it is unknown whether, say, Super Mario World is in NP. We will summarize exactly what is known in the table below. If readers have additions for newer games (for instance, it’s plausible that Super Mario Galaxy could be adapted to fit the same construction as the original Super Mario Bros.), please leave a comment with justification and we can update the table appropriately. I admit my own unfamiliarity with some of the more recent games.

Super Mario Brothers:

Game Title NP-hard in NP PSPACE-complete
Super Mario Bros.
Lost Levels
Super Mario Bros. 2
Super Mario Bros. 3
Super Mario Land
Super Mario World
Land 2: 6 Golden Coins
Super Mario Land 3
Yoshi’s Island
Super Mario 64
Sunshine
New Super Mario Bros.
Galaxy
New Super Mario Bros. Wii
Galaxy 2
Super Mario 3D Land

Legend of Zelda:

Game Title NP-hard in NP PSPACE-complete
The Legend of Zelda
A Link to the Past
Ocarina of Time
Oracle of Seasons
Oracle of Ages
The Wind Waker
The Minish Cap
Twilight Princess
Phantom Hourglass
Spirit Tracks
Skyward Sword

Donkey Kong:

Game Title NP-hard in NP PSPACE-complete
Donkey Kong
Donkey Kong Country
Donkey Kong Land
Country 2: Diddy’s Kong Quest
Land 2
Country 3: Dixie Kong’s Double trouble!
Land 3
Donkey Kong 64
Donkey Kong Country Returns

Metroid:

Game Title NP-hard in NP PSPACE-complete
Metroid
Metroid II: Return of Samus
Super Metroid
Fusion
Prime
Prime 2: Echoes
Prime Hunters
Prime 3: Corruption
Other M

Pokemon:

Game Title NP-hard in NP PSPACE-complete
All Games
Only trainers