# Haskell Quiz/Happy Numbers/Solution Dolio

The important thing to know is that there is only one eventual infinite cycle other than 1 => 1, namely:

- 4 => 16 => 37 => 58 => 89 => 145 => 42 => 20 => 4

Every positive integer is either happy, or eventually reaches that cycle, so one can arbitrarily choose one of those numbers to terminate a search and decide that a number is unhappy.

This is an easy job for memoizing/dynamic programming. The code specific to the problem looks like so:

```
module Main where
import Data.List
import Control.Arrow
import Memoizing
square x = x * x
comparing f a b = compare (f a) (f b)
digits = unfoldr mdiv
where mdiv 0 = Nothing
mdiv n = Just (m, d)
where (d, m) = divMod n 10
happy :: Integer -> (Bool, Integer)
happy = dpm happy'
where
happy' f 1 = (True, -1)
happy' f 4 = (False, -1)
happy' f n = second (+1) . f . sum . map square . digits $ n
happiest n = head . sortBy (flip . comparing $ snd . snd)
. filter (fst . snd)
. map (\n -> (n, happy n)) $ [1..n]
```

I arbitrarily picked 4 to terminate the unhappiness search. happy returns both the happy/unhappy status of the given number, and what order happiness it has (as defined on the rubyquiz page). happiest finds the highest order happy number between 1 and n.

The rest is a fairly reusable (for this sort of problem) module for dynamic programming (influenced by the memoizing recursion article on the old wiki; the original mailing list post by Chris Okasaki that inspired it all is here: http://www.haskell.org//pipermail/haskell-cafe/2005-July/010714.html):

```
{-# OPTIONS -fglasgow-exts -fallow-overlapping-instances #-}
module Memoizing(dp, dpm) where
import Data.Array
import Data.Bits
import Prelude hiding (lookup)
tabulate :: (Ix a) => (a,a) -> (a -> b) -> Array a b
tabulate bounds f = array bounds [(i,f i) | i <- range bounds]
-- Array-based, bounded dynamic programming. dp will take an upper and lower
-- bound, and memoize a function between those bounds
dp :: (Ix a) => (a,a) -> ((a->b) -> a -> b) -> a -> b
dp bounds f = (memo!) where
memo = tabulate bounds (f (memo!))
-- A type class for a memoizing map
-- m is the map type
-- k is the key type
-- v is the value type
-- fromFunction should build up a (possibly infinite) map for all keys, where
-- any key is mapped to the value of the function at that key.
class DPMap m k v | k v -> m where
fromFunction :: (k -> v) -> m
lookup :: k -> m -> v
-- dpm uses the above DPMap class to memoize functions with a potentially
-- unbounded domain
dpm :: (DPMap m k v) => ((k -> v) -> k -> v) -> k -> v
dpm f = memoized
where
m = fromFunction (f memoized)
memoized = flip lookup m
-- little-endian Patricia trees
data Patricia v = Nil | Node v (Patricia v) (Patricia v)
build n m f = Node (f n) (build n m' f) (build (setBit n m) m' f)
where m' = m+1
lookupP k (Node v l r)
| k == 0 = v
| not m = lookupP d l
| m = lookupP d r
where
d = shiftR k 1
m = testBit k 0
-- Patricia trees can be defined as DPMaps for keys of type (Bits k) => k,
-- however, I have defined them only for Int and Integer to save space
-- (one has to declare them all individually,
--
-- Bits k => DPMap (Patricia v) k v
--
-- won't work)
instance DPMap (Patricia v) Int v where
fromFunction = build 0 0
lookup = lookupP
-- DPMap instances for up to 4-tuples. In general, a map (k1,k2) -> v
-- is a map k1 -> (k2 -> v) and so on
instance DPMap (Patricia v) Integer v where
fromFunction = build 0 0
lookup = lookupP
instance (DPMap m1 k1 m2, DPMap m2 k2 v) => DPMap m1 (k1,k2) v where
fromFunction f = fromFunction (\i -> fromFunction (\j -> f (i,j)))
lookup (i,j) = lookup j . lookup i
instance (DPMap m1 k1 m2, DPMap m2 k2 m3, DPMap m3 k3 v) =>
DPMap m1 (k1,k2,k3) v where
fromFunction f = fromFunction (\i ->
fromFunction (\j ->
fromFunction (\k -> f (i,j,k))))
lookup (i,j,k) = lookup k . lookup j . lookup i
instance (DPMap m1 k1 m2, DPMap m2 k2 m3, DPMap m3 k3 m4, DPMap m4 k4 v) =>
DPMap m1 (k1,k2,k3,k4) v where
fromFunction f = fromFunction (\i ->
fromFunction (\j ->
fromFunction (\k ->
fromFunction (\l -> f (i,j,k,l)))))
lookup (i,j,k,l) = lookup l . lookup k . lookup j . lookup i
```