# Euler problems/151 to 160

## 1 Problem 151

Paper sheets of standard sizes: an expected-value problem.

Solution:

`problem_151 = undefined`

## 2 Problem 152

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.

Solution:

```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..79],
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, s) where x is a rational number whose reduced
-- denominator is not divisible by any prime greater than 3;
-- and s is all sets of numbers up to 80 divisible
-- by a prime greater than 3, whose sum of inverse squares is x.
only23 = foldl f [(0, [[]])] [13, 7, 5]
where
f a p = collect \$ [(y, u ++ v) | (x, s) <- a,
(y, v) <- pfree (terms p) x p,
u <- s]
terms p = [n * p | n <- [1..80`div`p],
all (\q -> n `mod` q /= 0) \$
11 : takeWhile (>= p) [13, 7, 5]
]
collect = map (\z -> (fst \$ head z, map snd z)) .
groupBy fstEq . sortBy cmpFst
fstEq  (x, _) (y, _) = x == y
cmpFst (x, _) (y, _) = compare x 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 80 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 <= 80]

solutions = if verify
then [sort \$ u ++ v | (x, s) <- only23,
u <- findInvSq (1%2 - x) all23,
v <- s]
else undefined

problem_152 = length solutions```

## 3 Problem 153

Investigating Gaussian Integers

Solution:

`problem_153 = undefined`

## 4 Problem 154

Exploring Pascal's pyramid.

Solution:

`problem_154 = undefined`

## 5 Problem 155

Counting Capacitor Circuits.

Solution:

`problem_155 = undefined`

## 6 Problem 156

Counting Digits

Solution:

`problem_156 = undefined`

## 7 Problem 157

Solving the diophantine equation 1/a+1/b= p/10n

Solution:

`problem_157 = undefined`

## 8 Problem 158

Exploring strings for which only one character comes lexicographically after its neighbour to the left.

Solution:

`problem_158 = undefined`

## 9 Problem 159

Digital root sums of factorisations.

Solution:

`problem_159 = undefined`

## 10 Problem 160

Factorial trailing digits

We use the following two facts:

Fact 1: (2^(d + 4*5^(d-1)) - 2^d) `mod` 10^d == 0

Fact 2: product [n | n <- [0..10^d], gcd n 10 == 1] `mod` 10^d == 1

Fact 1 follows from the fact that the group of invertible elements of the ring of integers modulo 5^d has 4*5^(d-1) elements.

Fact 2 follows from the fact that the group of invertible elements of the ring of integers modulo 10^d is isomorphic to the product of a cyclic group of order 2 and another cyclic group.

Solution:

```problem_160 = trailingFactorialDigits 5 (10^12)

trailingFactorialDigits d n = twos `times` odds
where
base = 10 ^ d
x `times` y = (x * y) `mod` base
multiply = foldl' times 1
x `toPower` n = multiply . genericTake n \$ genericReplicate n x
e = facFactors 2 n - facFactors 5 n
twos
| e <= d    = 2 `toPower` e
| otherwise = 2 `toPower` (d + (e - d) `mod` (4 * 5 ^ (d - 1)))
odds = multiply [odd | a <- takeWhile (<= n) \$ iterate (* 2) 1,
b <- takeWhile (<= n) \$ iterate (* 5) a,
odd <- [3, 5 .. n `div` b `mod` base],
odd `mod` 5 /= 0]

-- The number of factors of the prime p in n!
facFactors p = sum . zipWith (*) (iterate (\x -> p * x + 1) \$ 1) .
tail . radix p

-- The digits of n in base b representation
radix p = map snd . takeWhile (/= (0, 0)) .
iterate ((`divMod` p) . fst) . (`divMod` p)```