# Euler problems/171 to 180

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## Revision as of 08:42, 26 February 2008

## Contents |

## 1 Problem 171

Finding numbers for which the sum of the squares of the digits is a square.

Solution: this was Antoine Celeriers C code pasted from the Project Euler Forum.

## 2 Problem 172

Investigating numbers with few repeated digits.

Solution:

factorial n = product [1..toInteger n] fallingFactorial x n = product [x - fromInteger i | i <- [0..toInteger n - 1] ] choose n k = fallingFactorial n k `div` factorial k -- how many numbers can we get having d digits and p positions p172 0 _ = 0 p172 d p | p < 4 = d^p | otherwise = (p172' p) + p*(p172' (p-1)) + (choose p 2)*(p172' (p-2)) + (choose p 3)*(p172' (p-3)) where p172' = p172 (d-1) problem_172= (p172 10 18) * 9 `div` 10

## 3 Problem 173

Using up to one million tiles how many different "hollow" square laminae can be formed? Solution:

problem_173= let c=div (10^6) 4 xm=floor$sqrt $fromIntegral c k=[div c x|x<-[1..xm]] in sum k-(div (xm*(xm+1)) 2)

## 4 Problem 174

Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements.

## 5 Problem 175

Fractions involving the number of different ways a number can be expressed as a sum of powers of 2. Solution:

sternTree x 0=[] sternTree x y= m:sternTree y n where (m,n)=divMod x y findRat x y |odd l=take (l-1) k++[last k-1,1] |otherwise=k where k=sternTree x y l=length k p175 x y= init$foldl (++) "" [a++","| a<-map show $reverse $filter (/=0)$findRat x y] problems_175=p175 123456789 987654321 test=p175 13 17

## 6 Problem 176

Rectangular triangles that share a cathetus. Solution:

--k=47547 --2*k+1=95095 = 5*7*11*13*19 lst=[5,7,11,13,19] primes=[2,3,5,7,11] problem_176 = product[a^b|(a,b)<-zip primes (reverse n)] where la=div (last lst+1) 2 m=map (\x->div x 2)$init lst n=m++[la]

## 7 Problem 177

Integer angled Quadrilaterals.

Solution: This C++ solution is stolen from balakrishnan. Check out the forum if you want to see his solution to the problem

## 8 Problem 178

Step Numbers

Count pandigital step numbers.

import qualified Data.Map as M data StepState a = StepState { minDigit :: a , maxDigit :: a , lastDigit :: a } deriving (Show, Eq, Ord) isSolution (StepState i a _) = i == 0 && a == 9 neighborStates m s@(StepState i a n) = map (\x -> (x, M.findWithDefault 0 s m)) $ [StepState (min i (n - 1)) a (n - 1), StepState i (max a (n + 1)) (n + 1)] allStates = [StepState i a n | (i, a) <- range ((0, 0), (9, 9)), n <- range (i, a)] initialState = M.fromDistinctAscList [(StepState i i i, 1) | i <- [1..9]] stepState m = M.fromListWith (+) $ allStates >>= neighborStates m numSolutionsInMap = sum . map snd . filter (isSolution . fst) . M.toList numSolutionsOfSize n = sum . map numSolutionsInMap . take n $ iterate stepState initialState problem_178 = numSolutionsOfSize 40

## 9 Problem 179

Consecutive positive divisors.

## 10 Problem 180

Rational zeros of a function of three variables. Solution:

import Data.Ratio {- After some algebra, we find: f1 n x y z = x^(n+1) + y^(n+1) - z^(n+1) f2 n x y z = (x*y + y*z + z*x) * ( x^(n-1) + y^(n-1) - z^(n-1) ) f3 n x y z = x*y*z*( x^(n-2) + y^(n-2) - z^(n-2) ) f n x y z = f1 n x y z + f2 n x y z - f3 n x y z f n x y z = (x+y+z) * (x^n+y^n-z^n) Now the hard part comes in realizing that n can be negative. Thanks to Fermat, we only need examine the cases n = [-2, -1, 1, 2] Which leads to: f(-2) z = xy/sqrt(x^2 + y^2) f(-1) z = xy/(x+y) f(1) z = x+y f(2) z = sqrt(x^2 + y^2) -} unique :: Eq(a) => [a] -> [a] unique [] = [] unique (x:xs) | elem x xs = unique xs | otherwise = x : unique xs -- Not quite correct, but I don't care about the zeros ratSqrt :: Rational -> Rational ratSqrt x = let a = floor $ sqrt $ fromIntegral $ numerator x b = floor $ sqrt $ fromIntegral $ denominator x c = (a%b) * (a%b) in if x == c then (a%b) else 0 -- Not quite correct, but I don't care about the zeros reciprocal :: Rational -> Rational reciprocal x | x == 0 = 0 | otherwise = denominator x % numerator x problem_180 = let order = 35 range :: [Rational] range = unique [ (a%b) | b <- [1..order], a <- [1..(b-1)] ] fm2,fm1,f1,f2 :: [[Rational]] fm2 = [[x,y,z] | x<-range, y<-range, let z = x*y * reciprocal (ratSqrt(x*x+y*y)), elem z range] fm1 = [[x,y,z] | x<-range, y<-range, let z = x*y * reciprocal (x+y), elem z range] f1 = [[x,y,z] | x<-range, y<-range, let z = (x+y), elem z range] f2 = [[x,y,z] | x<-range, y<-range, let z = ratSqrt(x*x+y*y), elem z range] result = sum $ unique $ map (\x -> sum x) (fm2++fm1++f1++f2) in (numerator result + denominator result)