Combinatris: How to Play

Dirk van Deun, dirk at dinf.vub.ac.be

Combinatris is played with four combinators, represented by the letters I, K, S and Y. These appear at the left side of the playing field and move to the right. You use the up and down arrow keys to distribute them over the five available lines. This way you build five combinations at the right side of the playing field. Each time you add an extra combinator, the resulting combination will be reduced automatically according to the rules below. The game ends when one of your combinations fills a whole line and cannot be reduced.


Which reduction rule applies is determined by the leftmost combinator. The rule for combinations starting with K, for instance, says that you keep the first combinator that follows it, but not the next; and the K itself is "used up" during the reduction. Example 1: KSI reduces to S. Example 2: KS cannot be reduced: K is followed by only one other combinator, so the rule cannot be applied. Example 3: KSKI reduces to SI: the I at the end is not involved. As it does not matter to the rule what the second and the third combinator are, we can write the rule as Kxy → x. So here are the four rules:

Ix → x
The I combinator is pretty harmless by itself. An I combinator added to another combination just disappears.

Kxy → x
The K combinator is the one you want to get often. It shortens combinations as explained above, and one K at just the right position in a combination with lots of obnoxious S combinators can make the combination implode in a very satisfying way when you add the last S.

Sxyz → xz(yz)
The S combinator is the one you actually get often (and more and more often as you reach higher levels). The S combinator rule needs three more combinators to work: it rearranges them as shown, and you will have noticed that it introduces parentheses. So even though it is said to reduce the combination, it actually makes it longer: for instance, SSKI becomes SI(KI). This ugly rule was not made up as arbitrarily as it may seem: there is symmetry to it if you think of it as Sxyz → (xz)(yz). The first pair of parentheses are never written, however, as they would be dropped immediately anyway.

A pair of matching parentheses and everything in between counts as one big combinator. If such a composite combinator is in leftmost position however, its parentheses disappear. For example: K(IK)S reduces to (IK), which becomes IK as the parentheses are dropped, so the combination can reduce further to K.

Yx → x(Yx)
Y is the most powerful and also the most dangerous combinator, because reducing a combination that starts with Y immediately introduces a new Y. For instance, according to this rule, YI becomes I(YI); after which, according to the I rule, I(YI) reduces to (YI), which is simplified to YI by dropping the parentheses, which brings us back to our point of departure; which means that the whole thing can start over again. This could go on and on and bring in points without end, but it doesn't. If the game detects such an endless loop going on, it will end and you get no more points at all.

A combination with a Y combinator can also keep expanding endlessly, which is why the game will immediately end if a combination grows wider than the playing field, even if the combination can still be reduced.

Getting better at the game includes learning to recognize dangerous patterns (like YI) as well as common useful ones (like SK and YK) without having to think them through.


Parentheses do not only occur in the game as the result of S and Y reductions: they also appear as new elements for you to find a good spot for. They then come as matched pairs with two or three spaces in between. Combinators that bump into such pairs of parentheses will slide into them, until there is no room left in between. Other pairs of parentheses will too. Combinations with parentheses that still contain spaces are incomplete and can never be reduced. That is a good thing if you are building a long and complex combination and you do not want it to start reducing yet. It is a bad thing if you have too many of them and you cannot get rid of them fast enough.

Why you would build long and complex combinations on purpose brings us to scoring and strategy. Each reduction is good for 10 points, and so is the elimination of a pair of parentheses. For every 500 points you will go up one level, and the game will get harder, until you reach level 5. The challenge for the novice player is to reach level 5 (i.e. 2000 points); the challenge for the advanced player is to reserve one or two lines during the easier levels for building long and complex combinations, in such a manner that adding one extra combinator during level 5 can set off a chain reaction of reductions, yielding hundreds of points in one stroke during this otherwise almost unplayable level. (Hint: Y(SI) brings in many more points than Y.)

This makes the earlier levels harder, but it can give you a boost of about a thousand points at level 5. After that, even an experienced player will usually not last much longer. So for an advanced player, a score of 3000 is quite doable, but 4000 points is still a challenge.


When you have already moved a new combinator or pair of parentheses to the line where you want them, you can use the right arrow key to speed things up. The space bar will pause the game. Pausing the game to take some time to think should be considered cheating during a competition.

You might wonder why the letters I, K, S and Y were chosen as names for the combinators. They are taken from combinatory logic, which is a branch of mathematics. Our combinators I, K and S are exactly the same as in combinatory logic; but our Y combinator cheats a bit in the interest of gameplay, as in combinatory logic Yx is indeed equivalent with x(Yx), but does not directly reduce to it. (Our Y is in fact the similar but much less famous theta combinator.)


Play the game