Euler problems/151 to 160
Revision as of 13:52, 20 September 2007
Paper sheets of standard sizes: an expected-value problem.
problem_151 = undefined
Writing 1/2 as a sum of inverse squares
Note that if p is an odd prime, the sum of inverse squares of all terms divisible by p must have reduced denominator not divisible by p.
import Data.Ratio import Data.List invSq n = 1 % (n * n) sumInvSq = sum . map invSq subsets (x:xs) = let s = subsets xs in s ++ map (x :) s subsets _ = [] primes = 2 : 3 : 7 : [p | p <- [11, 13..83], all (\q -> p `mod` q /= 0) [3, 5, 7]] -- All subsets whose sum of inverse squares, -- when added to x, does not contain a factor of p pfree s x p = [(y, t) | t <- subsets s, let y = x + sumInvSq t, denominator y `mod` p /= 0] -- Verify that we need not consider terms divisible by 11, or by any -- prime greater than 13. Nor need we consider any term divisible -- by 25, 27, 32, or 49. verify = all (\p -> null $ tail $ pfree [p, 2*p..85] 0 p) $ 11 : dropWhile (< 17) primes ++ [25, 27, 32, 49] -- All pairs (x, n) where x is a rational number whose reduced -- denominator is not divisible by any prime greater than 3; -- and n>0 is the number of sets of numbers up to 85 divisible -- by a prime greater than 3, whose sum of inverse squares is x. only23 = foldl f [(0, 1)] [13, 7, 5] where f x p = collect $ concatMap (g p) x g p (x, n) = map (\(a, b) -> (a, n * length b)) $ pfree (terms p) x p terms p = [n * p | n <- [1..85`div`p], all (\q -> n `mod` q /= 0) [5, 7, 11, 13, 17]] collect = map (\z -> (fst $ head z, sum $ map snd z)) . groupBy cmpFst . sort cmpFst x y = fst x == fst y -- All subsets (of an ordered set) whose sum of inverse squares is x findInvSq x y = f x $ zip3 y (map invSq y) (map sumInvSq $ init $ tails y) where f 0 _ = [] f x ((n, r, s):ns) | r > x = f x ns | s < x =  | otherwise = map (n :) (f (x - r) ns) ++ f x ns f _ _ =  -- All numbers up to 85 that are divisible only by the primes -- 2 and 3 and are not divisible by 32 or 27. all23 = [n | a <- [0..4], b <- [0..2], let n = 2^a * 3^b, n <= 85] problem_152 = if verify then sum [n * length (findInvSq (1%2 - x) all23) | (x, n) <- only23] else undefined
Investigating Gaussian Integers
problem_153 = undefined
Exploring Pascal's pyramid.
problem_154 = undefined
Counting Capacitor Circuits.
problem_155 = undefined
problem_156 = undefined
Solving the diophantine equation 1/a+1/b= p/10n
problem_157 = undefined
Exploring strings for which only one character comes lexicographically after its neighbour to the left.
problem_158 = undefined
Digital root sums of factorisations.
problem_159 = undefined
10 Problem 160
Factorial trailing digits
problem_160 = undefined