Therefore, I will follow the excellent advice of Stephen Diehl, and will not write a monad-analogy tutorial. Instead, I will say: do the 20 Intermediate Exercises, and take the Monad Challenges. To paraphrase Euclid, there is no royal road to Monads; the exercises and challenges take you down that road at whose end you'll see at least part of why Monads are so useful.
If you were trying to learn group theory and people were standing around you saying "groups are like the integers", "groups are like Rubik's Cube", "groups are like m x n matrices", "groups are like baryons and mesons", your situation would be much like the student of Haskell amidst all the monad analogy tutorials. In a sense they're backwards. All those things are groups, just as burritos et al. can at least be thought…
I came across the excellent Monad Challenges, a collection of exercises designed to take you through what have become some of the standard example monads and take you through the process that motivates monads as a way to handle these seemingly diverse data structures. Both the Monad Challenges and the 20 Intermediate Exercises are well worth your time. Doing them both is helping me a lot.
That said, they don't quite match up. 20IE's banana is flip bind and apple is flip ap. (This isn't unique; the also excellent Learning Haskell from first principles has as an exercise writing bind in terms of fmap and join, but the declaration it gives has the type of flip bind. The authors do point this out.) As a result, I find myself with something that there's got to be some way to simplify, of the form
foo = flip $ (flip mumble) . (flip frotz)
I'd like to think there's some sort of distributive-like law to be found and used here... watch this space.
fivethirtyeight.com has a weekly column, "The Riddler". The Riddler poses two problems in each column. The first, "Riddler Express", is intended to be an easier problem that one can solve quickly, while the second, "Riddler Classic", is more difficult. Modulo vacations, it appears each Friday, and if you submit a solution by the end of the following Sunday (Eastern Time) , you will be among those who might be given credit for the solution in the next column.
The August 28th column's Classic problem is as follows: what's the longest sequence of integers x[i] for 1 ≤ i ≤ n such that for all i, 1 ≤ x[i] ≤ 100for all i < n, either x[i+1] is a multiple of x[i] or x[i+1] is a factor of x[i]for all i, j, x[i] = x[j] iff i = j, i.e. the x[i] are all distinct
(The last constraint avoids trivial sequences like 2, 4, 2, 4, 2, 4... which can go on forever if repeats are permitted.)
One way to characterize this problem is to look at it as a graph with a hu…