Alright, moving on with the representation series! This time I’ll start with an old puzzle that I, by coincidence, got from Steve Easterbrook and, separately, from Angelika Mader in Dagstuhl with a couple of weeks’ difference.
We have an 8 by 8 grid such as the one in the picture. We also have rectangular tiles that cover two adjacent squares of the grid. First question: can we completely cover the grid using these tiles? It should be easy to see that yes, we can do that quite easily.
But let’s make the puzzle a bit more complicated: we’ll take away the squares at two opposite corners of the grid, leaving 62 squares instead of 64, as shown below. Second question: Can we completely cover this grid? If so, how? If not, why not? (No overlaps, and no putting tiles out of the grid!)
The answer may not be obvious this time. We can try to work out a solution only to see it fail in the end:
(I’ll go over the solution now, so stop here if you wan to work it out yourself!)
Let’s substitute the 8 by 8 grid with (surprise!) a chessboard. If you take away two opposite corner squares, you’ll end up with a shape like this:
Note how the two removed squares are of the same color. In this case we have 32 remaining whites and 30 remaining blacks. And since our rectangular tiles each cover 1 black and 1 white cells, we will only be able to cover, at most, 30 and 30. There will always be two squares of the same colour remaining, so there is no way to arrange our tiles in the grid.
The answer to this puzzle is quite elegant. It’s so nice that it works not only with 8 by 8 grids, but with square grids with even lengths of any size (including, say, a million by million grid!).
What is it about this second perspective on the problem that makes its solution so trivial? It avoids the issue with the first: problem size. With the first perspective (where we’re putting tiles in the grid to try and stumble upon a solution) there are so many permutations that it’s hard to see whether we’ve tried them all or, perhaps, the real solution is still hiding somewhere, and we just need one more try to find it.
Re-representing the problem in terms of number of blacks and number of whites avoids the size problem entirely. We don’t need to try out every permutation, we don’t need to try out even one.
I have mentioned before how defining whether a representation is useful or not depends on the tasks it is needed for, and on the context of its reader. Now I’m adding a third consideration: the usefulness of a representation depends on how it handles the size of the problem it is representing. Some scale up pretty well, some others don’t, and for large problems the difference is key.
Two more examples of problem-size issues before I’m gone. Both have to do with chaotic libraries.
In the book “How Would You Move Mount Fuji?”, William Poundstone describes several puzzles that high-tech companies, particularly Microsoft, like to ask to their hiring candidates. Apparently they like to ask, among others, the following: How would you locate a specific book in a big library? There’s no cataloguing system and no librarian to help you.
Poundstone explains: “Suppose the books are in random order, which they might be, for all you know. In that case, the best you can do is to scan the shelves methodically (…) On the average, you would expect to have to scan half the library to find a given book.” Ouch!
However, he continues, it is perfectly reasonable to expect that the library will have an order, any order, straightforward or bizarre. What we should do in such a situation, then, is to map out the library and try to detect patterns in the types of books in each shelf. “The best approach is to first try to learn the system, then use that system to direct your search for the book you want.” In other words, we need to transform our perspective; to go from a representation where size is all important (needle in a haystack), to one where size becomes almost irrelevant. Puzzle-solving, in particular, frequently depends on this type of transformation.
The most deliciously complex library that I have ever read about is Jorge Luis Borges’ “Library of Babel”; an “indefinite, perhaps infinite” collection of rooms, shelves, and books. The Library has existed forever, and there is nothing outside of it. Each of its books has a seemingly random sequence of letters, and some of this Universe’s inhabitants have conjectured that there are books for every possible permutation of characters, meaningful or not –every story and non-story can be found in it. At one point, the narrator mentions a superstition some librarians have: that someone, somewhere, has deciphered the order of the Library, that “there must exist a book which is the formula and perfect compendium of all the rest: some librarian has gone through it and he is analogous to a god.” Hereticals, on the other hand, “maintain that nonsense is normal in the Library and that the reasonable (and even humble and pure coherence) is an almost miraculous exception”.
I’d say you must read this short story if you haven’t. You can find its full text here.