How to solve combinatorial puzzles (and don't get puzzled)

From time to time, someone is asking me for a help to solve some puzzle. It is often inspiring matter for me because I wouldn't even dream of such problems that are tried to be solved by the questioners. I am happy for them. They are giving me an opportunity to broaden my horizons and look at it also from the other site, which wouldn't be otherwise considered.

Sometimes the problem is caused by a negligence - who would like to read lengthy instructions if only two pieces on a puzzle need to be swapped, right? Sometimes a mistake is on my side - the tutorial is either not understandable or written not explicitly enough. And there are even such cases in which a tutorial is not applicable at all because it can not be followed in terms of solving process (for whatever reason).

Once in a while I manage to solve some puzzle by accident. Here are some not obligatory tips (almost tricks) I am trying to obey while solving. I believe you might find them helpful, too.

Solve already before scrambling

Perhaps you are saying to yourself: "Such a stupidity for the first advice". The purpose of this verbal paradox should be to point out that when you completely scramble the puzzle right after handling it for the first time, you will possibly solve it much harder in comparison with a proper inspection of a solved puzzle before scrambling. Personally, I even document it sometimes by taking a picture of all puzzle sides.

Since you are probably not going to see a brand new puzzle to be solved (for some time) after its scrambling, a documentation may come in handy. From my experience, I can confirm there is nothing worse than if you know how to solve the puzzle but you don't know how actually a solved state looks like (so basically you are never sure whether the puzzle is solved or not).

To some extent, a simulator of given puzzle can replace a documentation. However, not all puzzles are simulated yet and not everyone can program it if needed. Still, a computer simulator is irreplaceable in cases when we want to try some moves on a physical puzzle without the risk of scrambling it.

Less is more

I hear you saying: "Such a stupidity for the second advice". But sometimes less can be more indeed, and in case of a solution to combinatorial puzzles it's especially true.

When human being encounters a certain problem, he/she wants to solve it at once. That's probably given by nature. Consider an ordinary purchase of apples and pears as a problem. You come into the store, buy apples and pears, and go home. Surely you will agree that coming into the store, buying pears only, going home and then repeating a process for buying only the apples would be inefficient.

This logic, however, doesn't always work for the puzzles - at least in terms of intuitive solving. If you would like to solve more types of different pieces on the puzzle at once (which is an analogy to buying apples and pears simultaneously), it can admittedly be more efficient than solving one piece type at a time. Nevertheless, as a side effect, you will mostly be forced to find new algorithms, thus you won't be able to continue in intuitive way of solving.

I think the more puzzles you can solve, the less algorithms you want to memorize for each of them. There are even puzzles having more than 4 different piece types. For those, if you want to solve even just 2 piece types at once, it is required to remember hundreds, if not thousands (in extreme cases much, MUCH more) algorithms.

So ask yourself now: is it more profitable to solve a puzzle faster and perhaps by fewer moves using a lot of algorithms (i.e. complexly), or is it better to solve it slowly but for certain - without algorithms (i.e. easily)? For those of you who have been chosen the latter option, I have another wisdom - I am convinced that in simplicity is not only a power, but also an elegance!


The abbreviation CCC (KKK in Czech) can stand for whatever. From Ku-Klux-Klan to Commutators + Conjugations Combination. The purpose of CCC is to solve only a few pieces without unnecessary scrambling of already solved parts. If you can come up with your own commutator during a solve, you basically won. It doesn't have to be only a commutator, generally it can be any algorithm that solves not so many pieces. It may not be easy at all to find one, though. Once you succeed, it would be a pity to not use it for further solving. By setup moves, application of algorithm and inverse setup moves it is usually possible to completely solve a certain puzzle piece type.

Take notes

When you see scrambled puzzle in front of you, try to set a goal (rather modest than colossal one) you want to achieve - cycling of three pieces, for example. Try doing move after move on a puzzle to meet your goal (other types of pieces don't have to be considered for now). Maybe you won't succeed for the first or tenth time, not even for the hundredth time. In that case, a change of solving strategy is worthy. Experiment. It will pay off (ahem, sometimes). Speaking of which, Mr. Graham Parker from England has been apparently solving the 3x3x3 Rubik's cube for incredible 26 years. But he succeeded!

Once you reach your goal, take a pencil with a paper and write down the procedure before you forget it. So what that my doodles sometimes look like a message encrypted by Enigma? In addition to the letters and numbers, on my sketch can be often found arrows, double-arrows, straight and rounded arrows, as well as asterisks, squares, commas and generally everything what can be used for the best description of the procedure.

Since you rarely manage to solve the puzzle in one sitting, you will appreciate your notes with gratitude later, when you come back to a puzzle. You will be able to continue exactly where you left off. Right after your decryption of your notes. It is not recommended to use too many symbols for your notation. Believe that the greater is a time interval between taking them and looking at them again (and the more symbols you will use), the longer time will be required to spend in order to re-understand them.

Pause - first thing to do

I've heard somewhere that the human brain is capable of working for 10, at most 15 minutes in a state of complete body concentration. There's something about it. Honestly, think of your school years - how long did you pay attention during teacher's explanation, even if the topic was interesting for you?

If you don't succeed to achieve your goal on a puzzle in half an hour, I warmly recommend to go somewhere else and make your mind clear. Try jogging, cutting the grass or cleaning up the house. Any activity, preferably physical one, that helps you to forget about the puzzle for a while will probably come in handy.

It is quite possible you will still subconsciously think of a puzzle, even if your mind is almost fully occupied with your activity. Right at this moment, the time for a birth of a so-called "AHA moment" has come. AHA moment is simply a moment when you suddenly realize some ground-breaking idea that has been eluding you so far (maybe just because you spent a lot of time with a puzzle, thus making your brain "dull").

Aha moment is one thing, its application is another thing - unfortunately, AHA moment has often some side effects that won't simply occur to you without having a real puzzle with you. Nevertheless, there is always the possibility to change a solving technique if needed, or a chance for the birth of new AHA moment that will solve previous troubles. I am not sure why but the best AHA moments occur to me when I am in the bathtub. History, however, suggests that this is not quite as unique situation - remember Archimedes (like he couldn't have been yelling "AHA", right?).

AHA moment may come anytime - i.e. between a minute and let's say 26 years. Usually the body itself will tell whether it wants to return to the puzzle, eventually for how long. If body says no and you have a resistance to further solving, it is quite possible that the AHA moment will simply never occur to you.

It is good to notice that the basis of the word "brainteaser" (puzzle) is to tease your brain. If you don't want to tease your brain anymore, that's fine - nothing happens (as it's not a question of life or death). You won't be the first nor the last person who fails to solve the puzzle (at least without a tutorial). Pushing yourself into something you don't want to do makes no sense. Certainly don't go as far so that the puzzle would hunt you and make you mentally mad. You are the master of it, it is not master of yours.

On contrary, if you do want to tease your brain, the puzzle should be an entertainment or relaxation for you, to some extent. In such case you shouldn't be solving it just because you are supposed to by someone, instead, you should be solving it because you want to do it and because you find it really enjoyable. Let the pleasure of finding things out be your reward, at least.

Nothing like a known procedure

Right now you won't probably tell what is equal to 89720567·3416716. You have learned how to calculate this product once in a past - the principle remains the same as for a product of 2·3. Let's apply a math problem (product of two numbers) to a solution to combinatorial puzzles.

Just as there are many fundamentally different ways (you met first of them in the third grade, I guess) how to calculate a given product of 89720567·3416716, there are also several methods (techniques, strategies or approaches, if you like) how to solve combinatorial puzzle. For instance, if you can solve a 3x3x3 Rubik's cube (let's replace it by the product of 89720567·3416716 for a moment), maybe you don't know you are also able to solve a 2x2x2 Rubik's cube (which represents the product of 2·3) using a similar procedure.

Therefore, it is useless to look at two different-looking puzzles as being two independent puzzles, if we can (partially, at least) mathematically describe them by the same laws - see an analogy to the products of 89720567·3416716 and 2·3.

So, when solving puzzles, maximally use the knowledge from the past (the more puzzles you have solved in the past, the better for you) - by the way, it corresponds nicely with a concept of CCC (see above). Sometimes it is difficult to figure the laws out at first glance (both in mathematics and in puzzles - the one who has been trying to solve e.g. the Fisher cube and 3x3x3 Rubik's cube knows what I mean), but that's probably why a puzzle is being called a puzzle.

(Don't) cheat!

One of the most unpleasant things that can happen to you when solving is if you get already scrambled puzzle in your hand. If you can "reduce" it to some other puzzle which you can solve (you know a solution to it from the past), that's the better option. But if that scrambled puzzle is completely unknown to you, it may raise a real problem. While solving this problem, even a tutorial won't be helpful, since it will be unusable.

So what is meant by a real problem? Aside from not knowing how a solved puzzle looks like (you can usually handle it by figuring out its color scheme), the real problem is that the puzzle can be unsolvable at this point. At least by doing those moves that can be normally applied to the puzzle (relabeling of stickers and mechanical disassembly + reassembly doesn't count here).

So what is causing puzzle unsolvability? Well, the causes were just said - either someone (typically four years old younger brother) could re-label stickers or the puzzle could be disassembled and reassembled again. However, while doing so, an emphasis didn't have to be put on whether all puzzle pieces are further solvable (which is usually not met in case of getting all pieces back randomly). The reason for puzzle disassembly can be, for instance, a desire to see inner mechanism (some of them are truly fascinating and impressive) or a so-called POP - flying a piece out of the puzzle as a result of violation of its alignment and attempt to execute next move (typically in case of speedsolving).

By trying to solve such unsolvable puzzle you can spend your youth (in the best case). Based on my personal experience, it turned out that if I get beforehand scrambled puzzle and can't figure it out for a long time no matter what, it is better to disassemble it into particular parts (that would be considered as cheating under normal circumstances) and reassemble them back mechanically in a way so that the puzzle would be solved. It is the only way to be sure that the puzzle will be always solvable, unless you make it unsolvable again.

Google is your friend

As in almost everything, even when solving combinatorial puzzles, you can choose from at least two options. The first one is difficult, full of obstacles and unexpected traps. It is represented by intuitive solving. The latter is represented by comfy (often mindless) execution of moves which were found for us by someone else - either a human or an artificial intelligence.

Nobody is perfect and everyone occasionally needs a little kick (not butt-kick, though) in order to solve a certain problem - puzzle in our case. There is nothing wrong on using all available ways and searching for information on the internet. Therefore, the following section will deal with other, less "challenging", solution options.

I think it is quite unlikely that the puzzle you are trying to solve would remain without someone else's attention who already tried to solve it before. Odds are someone did it and he/she shared his/her achievement with a world.

Over the last few years (written in March 2014), there was a rapid increase of websites where one can find solutions to combinatorial puzzles. Just type a name of the puzzle into Google and browse the first few found results. You can't go wrong by specifying the query - just write something like "tutorial" or "solution" after the puzzle name.

Not every solver, however, wants to make a website. Instead of Google, connect to YouTube (Google will probably force you to it through search results anyway) and look around there.

Not every solver, however, wants to create a video. If Google and YouTube fail, yet you still want to know a tutorial for your puzzle, there are specialized discussion forums for you. The members of those are usually not only questioners/candidates for advice, but also puzzle designers + builders (for real as well as virtual puzzles), successful puzzle solvers (which you will surely appreciate), mathematicians and simply all people who are hanging around the puzzles. I haven't experienced a situation in which I wouldn't find a tutorial for wanted combinatorial puzzle in some form on the forum. If this happens to you, simply start a new thread and ask for an advice. Write where you got stuck or what is troubling you. Since generally it is not easy to describe a situation on the puzzle in detail, I recommend (in case you don't know notation) to enclose also a picture (photo or sketch for a real puzzle, printscreen for a virtual puzzle) to your post. Ideally, that picture will show a situation on the puzzle (and why are you actually puzzled).

The disadvantage (for foreign speakers) of searching on the internet is a necessity to speak, at least partially, a foreign language (English will suffice for sure). For example, the few tutorials that exist in Czech are usually not of such quality as in case of English equivalent.

People talented in programming can further use a computer. Perhaps the easiest method is using brute force, i.e. searching a combination after combination in order to find a solution. However, it is of limited scope - if the number of combinations for a puzzle is greater than tens of millions, it is not possible to get a real-time solution (programmers, please correct me if necessary). In such a case, there is a need to either reduce the number of possible combinations (i.e. solve some puzzle pieces and run the program again), or write more sophisticated methods that will be able to solve such a large number of combinations in a real time.

„Na světě je zavedeno, že spousta hloupých si hraje na chytré. Z chytrých, kterých je na světě nedostatek, jen ti nejchytřejší si hrají na hloupé.“ Jan Werich