In a recent post on the SIAM blog, Professor Des Higham discusses cryptic crosswords and the reasons why they appeal to so many mathematicians of his acquaintance. I’m one of those mathematicians, having tried my hand both at solving and at compiling crosswords, and Des’s article got me thinking a little more about where their appeal really lies. Clearly there’s the fact that mathematicians, in general, simply like solving puzzles; indeed, the boundaries between recreational puzzles and “genuine” mathematics can sometimes become so blurred as to be non-existent, as in the early work on gambling or the Königsberg bridges problem. But can we, I wondered, go a bit further than this and look at crossword-solving through the lens of mathematical thinking, or vice versa?I think that one of Des’s statements, which is that “solutions are discovered through creative use of logical steps”, requires a little expansion. In my experience — which might be entirely atypical — the steps involved in discovering a solution are very far from logical: they involve vague associations, working hypotheses that are briefly considered and then discarded, and occasional outrageous guesses; and it is only after a solution has presented itself that formal reasoning steps in to check whether the solution is in fact valid. In this respect, of course, crosswords may have still more in common with mathematical research than Des suggests!
Perhaps the best way to express this is to say that what crossword solving shares with mathematics is the eerie power of a good heuristic. This shows up most obviously in anagrams. By my reckoning, there are 9! = 362880 ways to arrange the letters in a nine-letter anagram (not at all uncommon in an everyday grid) and 15! = 1307674368000 ways to arrange the letters in a fifteen-letter anagram (rarer but hardly unknown). Despite this, many solvers have had the experience of looking at a lengthy anagram and spotting the answer within a second or so. It seems barely plausible that the brain has had time to implement a brute-force strategy such as working through all the possible combinations of letters, or even (more plausibly) thumbing through a mental dictionary of nine- or fifteen-letter words and checking each for compliance with the clue. Rather, what seems to happen is that we spot certain combinations of letters (like a T, an I, an O and an N), and consider the working hypothesis that our solution ends in -TION: this gives us a massively reduced brute-force problem to solve, and because of the highly restricted set of letter combinations to be found in any natural language, the strategy works pretty frequently. (The hardest anagrams to spot tend to be those of words which contain unusual letter combinations and don’t contain common prefixes or suffices — the well-known anagram of CARTHORSE is a case in point.)
Paradoxically, the longer an anagram becomes the easier it can be to solve: with some of the late and irreplaceable Araucaria’s colossal quotation anagrams, the most successful solution strategy was simply to look at the pattern of letters and wait for a plausible sequence of words to present itself. (To take a very sub-Araucarian example, if the pattern of letters is (2, 2, 2, 3, 2, 2) and you have a suspicion that the crossword is themed around lines from Shakespeare, then a possibility will rapidly present itself and render the clue “i.e.?” comprehensible!) In a cryptic clue, of course, the problem is rendered even easier by the promise that the clue will contain, somewhere, a “straight” definition of the solution, so another effective strategy is to consider the possible definitions in turn, and for each one riffle through your mental thesaurus looking for plausible candidates for the anagram.
The reason why heuristics work, in crosswords as in mathematics, is that we are playing a game in a highly conventionalised world. This is obvious in mathematics, where there are formal rules that determine whether a particular step is admissible or not, and where, less formally, our developing mathematical maturity gives us a sense of which claims are credible and which methods could plausibly lead to particular results. (A nice illustration of these informal rules is given by Scott Aaronson’s Ten signs a claimed mathematical breakthrough is wrong.) Crosswords, less obviously, don’t just work within a restricted “language” (there are, after all, only so many ways to indicate that a clue contains an anagram) and a set of formal rules (usually the Ximenean rules); they also have a surprisingly restricted field of reference. It’s hard, in my experience, to score a reasonable percentage against the broadsheet prize crosswords without some vague knowledge of classical music and cricket (both of which have substantial vocabularies crammed with specialised and quite odd words), but I can testify that it’s perfectly possible to get by with very little knowledge of, say, association football or popular music recorded since about 1995. In a sense — and I don’t want to push the analogy too far — the odd areas of knowledge that help disproportionately with crosswords are a bit like the collection of mathematical techniques and tricks that each of us learns are disproportionately helpful solving problems in our own area of mathematics. (An example of such a trick from my own field, fluid dynamics, is that of looking for self-similar solutions; I’m sure my colleagues in other disciplines have their own favourites.)
Of course, the danger is that, without an injection of the unexpected, crosswords can become rather stereotyped, and I think the same applies to mathematics. A great compiler has what I think is best described as wit: the ability to startle the solver with a surprisingly topical reference or linguistic oddity (the famous anagram of BRITNEY SPEARS, though it’s a bit dated now, is a case in point), or simply to approach the task of clueing from an entirely unexpected angle — again, Araucaria provides some perfect examples. Similarly, the difference between a journeyman piece of mathematics and the work of a real master is that the former will tackle a standard problem very competently in a standard way, whereas the latter will deploy an idea or a technique from an unexpected source, or address a problem that nobody beforehand would have thought was natural or feasible to tackle.
That is the lasting joy of both crosswords and mathematics: not bypassing the heuristics and the conventional (or even algorithmic) elements, but mastering them enough to appreciate the wit that operates within that framework and occasionally dances beyond it. It’s the joy that one also finds in poetry and in music and, I suspect, in every other sphere where an apparently restrictive system of rules becomes the essential support for creativity.
PS: ORCHESTRA; TO BE OR NOT TO BE; PRESBYTERIANS. There: you can stop worrying now…