Favourite Puzzle (Syderia)

[jack]

This was an excellent question to ask me, but unfortunately, I'm completely blocked on an answer! But I think it's interesting anyway, I will come back with an actual answer after December if one occurs to me.

OK, my theory of puzzles. I think a quintessential puzzle is one where, when you guess the answer or it's pointed out, it's immediately obvious that it's right (else it's not a puzzle, it's just difficult), but it's not obvious in advance. And it's hard to quantify this, but it should not just be a matter of exhaustively trying possible answers, there should be different potential avenues and be able to make a leap to the right answer (eg. "find the prime factors of $largenumber" isn't really a puzzle).

A common sort of puzzle is riddle-puzzles where you need to relax some assumption which you made, but explicitly wasn't included in the statement of the puzzle. These are normally carefully constructed, but can happen in real research too, where some insight that "hey, what if we DIDN'T do this" suddenly makes everything else fall into place.

Until I wrote the previous paragraph I hadn't thought of this as a category, but I asked, are there quintessential puzzles which are not riddle-puzzles? And I guess, they tend to be ones where the answer is reached by a chain of reasoning where each step isn't that hard, but it's impossible to see the next step until you've done the previous one.

Another sort of puzzle that shows up especially amongst mathematicians is an open-ended puzzle. Like, if there's a puzzle "five people have to do X in less than forty seconds", mathematicians don't even hear "five" and "forty", they hear "if N people have to do X, what's the least time it's possible to do it in? provide an example and proof there's no smaller answer". But often it's the case that the question asked is a good place to start, and if you can solve that, it's a similar, harder problem to solve it generalised in various ways. Obviously any question _could_ be generalised like this, but only some actually get more interesting as you do so: sometimes all the generalisations are either the same, or impossible, or a list of special cases.

I really need some specific examples, but can't think of anything good. Anyone else suggest any?

But I'll end on one that I remember, at school, proving that moving a tower of hanoi from one needle to another takes 2^n-1 moves. It was one of the first times I worked something out for myself and wrote it all up with a proof and conclusion, at mum's suggestion.

Comments

[seekingferret]

I think by your definition, a jigsaw puzzle is a quintessential puzzle that is not a riddle-puzzle.

Mystery Hunt jargon analyzes puzzles by 'aha's, which I think are the same as your description of the answer being obviously correct in retrospect but not at first glance. Mystery Hunt aesthetics tends to value as 'elegant' puzzles where in retrospect the aha couldn't have possibly been anything else, and regard puzzles where the aha is obviously correct in retrospect, but only one of many possible obviously correct answer as 'inelegant'.

[gerald_duck]

One conveniently easy problem (perhaps not puzzle) which your description of an open-ended puzzle reminds me of is to prove there can only be five (3D) platonic solids. How many regular polygons can fit around a point with angles summing to less than 360° (so that it's a polyhedron, not a plane tesselation)? Answer: 3, 4 or 5 triangles, 3 squares or 3 pentagons. Which turns out to be exactly the platonic solids which do exist.

Puzzle 5 here is one of my most fondly remembered from when I was a kid, which might make it one of my favourites.

There's also the blue forehead puzzle in its various forms.

[simont]

OK, my theory of puzzles. I think a quintessential puzzle is one where, when you guess the answer or it's pointed out, it's immediately obvious that it's right

Hmm. I've put some thought before now into what makes a problem (by which I mean any old question that requires thought to work out the answer) into a puzzle. I've toyed with various definitions in various contexts, and I think the one you suggest here is one of them. Another is that the method of solution should be non-obvious: if you look at the question, can readily see a procedure for producing the answer, and now all you have to do is grind mechanically through that procedure, that disqualifies the problem from being a puzzle. (Though you have to tweak that definition a bit in the face of procedures which are too tedious or unpleasant for a human solver to actually bother with: the trivial algorithm of 'iterate over every possible element of {1,2,...,9}⁸¹ checking the clues' does not disqualify Sudoku from puzzlehood, and I'd argue that 'write it down as a matrix over GF(2) and invert it' doesn't disqualify Flip either.)

But I think my favourite definition is that a puzzle is a problem chosen for a pleasurable solving experience. I think both of the above properties contribute to the pleasure of puzzle-solving: it's precisely the fact that the solution process included both the head-scratching 'no idea even where to start' stage and the delighted recognition of what's clearly the right answer, because having felt the former makes the latter so much more satisfying. And this definition also reminds you that you need other important features, such as having about the right difficulty level, so that your intended audience will experience both the initial bafflement and the final satisfaction.

[obandsoller]

I agree very much with this: "chosen for a pleasurable solving experience". Except possibly with the word "pleasurable" for reasons similar to why fun is a four letter word

http://www.escapistmagazine.com/articles/view/video-games/issues/issue_65/381-Fun-is-a-Four-Letter-Word

[mair_in_grenderich]

I picked up a book of puzzles while visiting someone recently, and where I opened it it read something of the form,

Mr X gives his daugher 25 sweets (or some number), to divide among her Y siblings proportional to their age, but boys get (relatively) twice as many as girls. Everyone gets whole sweets. What age are the children?

It might also have supplied the age of one of them, I can't remember. In any case I couldn't be bothered to try and work it out, but flicked to the back, where I discovered that although the sweets were whole, the siblings were not: half-sisters and half-brothers only got half as many sweets, and only in this way was a whole-sweet solution possible.

I rather felt that was cheating on the part of the puzzle setter, personally.

[mair_in_grenderich]

ps do you like crosswords?

[syderia]

Thank you for your answer (I didn't mean to stump you).

Recursive reasoning was one of my favourites, back when I used to do hardcore maths. It's so simple as a concept, and you get to demonstrate some really powerful things with it.

[jack]

Thank you for asking! It was definitely a really interesting topic, even though I didn't have a specific answer.

I think I used to like doing hard puzzles when I was young, but the thing with maths I didn't quite get used to until a way through university is the pattern of "here's one hard thing, ok we solved it, now we take that technique and systematise it so we can use it EVERYWHERE!" I had to learn the faith that sometimes a solution required LOTS of hard steps, often being ones you wouldn't usually figure out yourself, but learn as a standard technique.