# Euler problems/171 to 180

### From HaskellWiki

< Euler problems(Difference between revisions)

(add problem 176) |
(add problem 172) |
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Solution: |
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<haskell> |
<haskell> |
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− | problem_172 = undefined |
+ | factorial n = product [1..toInteger n] |

+ | |||

+ | fallingFactorial x n = product [x - fromInteger i | i <- [0..toInteger n - 1] ] |
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+ | |||

+ | choose n k = fallingFactorial n k `div` factorial k |
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+ | |||

+ | -- how many numbers can we get having d digits and p positions |
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+ | p172 0 _ = 0 |
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+ | p172 d p |
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+ | | p < 4 = d^p |
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+ | | otherwise = |
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+ | (p172' p) + p*(p172' (p-1)) + (choose p 2)*(p172' (p-2)) + (choose p 3)*(p172' (p-3)) |
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+ | where |
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+ | p172' = p172 (d-1) |
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+ | |||

+ | problem_172= (p172 10 18) * 9 `div` 10 |
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</haskell> |
</haskell> |
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## Revision as of 03:08, 1 February 2008

## Contents |

## 1 Problem 171

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

Solution:

problem_171 = undefined

## 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.

Solution:

problem_174 = undefined

## 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:

problem_177 = undefined

## 8 Problem 178

Step Numbers Solution:

problem_178 = undefined

## 9 Problem 179

Consecutive positive divisors. Solution:

problem_179 = undefined

## 10 Problem 180

Solution:

problem_180 = undefined