Yesterday, to keep myself occupied on a subway ride out to Brooklyn, I grabbed an old paperback copy of George Pólya’s classic book “How to Solve It” – about the art and science of problem solving – a book I’d never quite gotten around to reading. On one page that I turned to at random, he used the example of trying to make a one-word anagram from “DRY OX TAIL IN REAR”. The idea here is to rearrange those letters in such a way that they form a single word – using each of the letters exactly once.
It seemed like a really hard problem to me, and I figured I might be able to get a good fifteen or twenty minutes – or more – of fun on the long subway ride, trying to work it out with pen and paper. I committed the short but mysterious phrase to memory, and put the book back into my jacket pocket. Then I took a pen out of my pants pocket, together with the piece of paper where I’d written the address in Brooklyn I was going to, and I wrote out the words neatly so that I would be able to study them properly as I planned my attack:
DRY OX TAIL IN REAR
Settling in to what promised to be a fun and challenging subway ride, I figured I would begin with some likely letter combinations. I started to write some letters below the phrase, and those first letters, as they appeared on the paper one after another, turned out to be the solution. Apparently while I was busy fussing with putting away the book, taking out pen and paper, copying over the five words, my subconscious mind had been merrily working away – without my knowledge or consent – solving the darned thing.
I was disappointed – in fact I felt like I’d been cheated. It wasn’t even as though I could really take credit for the solution. My plan had been to systematically apply some of Pólya’s fascinating ideas about problem solving, and to enjoy a little study of heuristic methods at work. Instead my hand had just written it all out, full stop. I was left with no idea of either the approach or the method, since my subconscious mind has an annoying habit of not telling my conscious mind what it is thinking.
Yet I suppose this was an instructive experience. We go about our daily lives thinking that we are consciously working out solutions to all sorts of problems. The reality is probably quite different – our subconscious selves are really figuring out most of it. We just show up at the end to claim all the credit and the glory. The truth is that we rely on the subconscious to add that little something extra – ordinarily we don’t even notice it happening. But every once in a while the crafty little devil tips his hand.
i found it, but i’m not telling….
oh, reveal it. I have a feeling it’s a chemical thing, using dioxi…
I’m still thinking of your post in which you said that confidence plays a big role in performance. Once you had solved the Saturday crossword once, you could always solve it thereafter. I’m scare of being stupid, so won’t even let my mind play with DRY OX TAIL IN REAR.
Now that I’ve posed the problem, I can’t just reveal the answer the very next day, because that wouldn’t be fair to anybody who has taken it on as a challenge and is still working on it.
I, too, found it. It was uncommonly easy.
Of course this sort of thing is extremely well known in the community of research mathematicians, and not least because of the extraordinarily vivid and compelling account of the phenomenon bequeathed to them by Jules Henri Poincare (1854-1912), one of the greatest mathematicians of all time.
In his wonderfully lucid book SCIENCE AND METHOD, Poincare included a chapter entitled “Mathematical Creation,” in which he supplies several detailed examples of major breakthroughs that he attributes to exactly the sort of unconscious mathematical activity you describe in this blog entry. Then he goes on to supply a rough theory of the phenomenon, which has seemed quite persuasive to later mathematicians who have had similar experiences.
I venture to suspect that, if you haven’t yet read Poincare’s essay, you will probably find it every bit as stimulating as Polya’s classic little book.
ok., i’m slow. now tell me.
my friend just gave me the answer. i feel slow, tho will try to recover.