# Euler problems/101 to 110

### From HaskellWiki

## Contents |

## 1 Problem 101

Investigate the optimum polynomial function to model the first k terms of a given sequence.

Solution:

problem_101 = undefined

## 2 Problem 102

For how many triangles in the text file does the interior contain the origin?

Solution:

problem_102 = undefined

## 3 Problem 103

Investigating sets with a special subset sum property.

Solution:

problem_103 = undefined

## 4 Problem 104

Finding Fibonacci numbers for which the first and last nine digits are pandigital.

Solution:

Very nice problem. I didnt realize you could deal with the precision problem. Therefore I used this identity to speed up the fibonacci calculation: f_(2*n+k) = f_k*(f_(n+1))^2 + 2*f_(k-1)*f_(n+1)*f_n + f_(k-2)*(f_n)^2

import Data.List import Data.Char fibos = rec 0 1 where rec a b = a:rec b (a+b) fibo_2nk n k = let fkm1 = fibo (k-1) fkm2 = fibo (k-2) fk = fkm1 + fkm2 fnp1 = fibo (n+1) fnp1sq = fnp1^2 fn = fibo n fnsq = fn^2 in fk*fnp1sq + 2*fkm1*fnp1*fn + fkm2*fnsq fibo x = let threshold = 30000 n = div x 3 k = n+mod x 3 in if x < threshold then fibos !! x else fibo_2nk n k findCandidates = rec 0 1 0 where m = 10^9 rec a b n = let continue = rec b (mod (a+b) m) (n+1) isBackPan a = (sort $ show a) == "123456789" in if isBackPan a then n:continue else continue search = let isFrontPan x = (sort $ take 9 $ show x) == "123456789" in map fst $ take 1 $ dropWhile (not.snd) $ zip findCandidates $ map (isFrontPan.fibo) findCandidates problem_104 = search

It took 8 sec on a 2.2Ghz machine.

The lesson I learned fom this challenge, is: know mathematical identities and exploit them. They allow you take short cuts. Normally you compute all previous fibonacci numbers to compute a random fibonacci number. Which has linear costs. The aforementioned identity builds the number not from its two predecessors but from 4 much smaller ones. This makes the algorithm logarithmic in its complexity. It really shines if you want to compute a random very large fibonacci number. f.i. the 10mio.th fibonacci number which is over 2mio characters long, took 20sec to compute on my 2.2ghz laptop.

## 5 Problem 105

Find the sum of the special sum sets in the file.

Solution:

problem_105 = undefined

## 6 Problem 106

Find the minimum number of comparisons needed to identify special sum sets.

Solution:

problem_106 = undefined

## 7 Problem 107

Determining the most efficient way to connect the network.

Solution:

problem_107 = undefined

## 8 Problem 108

Solving the Diophantine equation 1/x + 1/y = 1/n.

Solution:

import List primes=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73] series _ 1 =[[0]] series xs n =[x:ps|x<-xs,ps<-series [0..x] (n-1) ] distinct=product. map (+1) sumpri x=product $map (\(x,y)->x^y)$zip primes x prob x y =head$sort[(sumpri m ,m)|m<-series [1..3] x,(>y)$distinct$map (*2) m] problem_108=prob 7 2000

## 9 Problem 109

How many distinct ways can a player checkout in the game of darts with a score of less than 100?

Solution:

problem_109 = undefined

## 10 Problem 110

Find an efficient algorithm to analyse the number of solutions of the equation 1/x + 1/y = 1/n.

Solution:

-- prob in problem_108 problem_110 = prob 13 (8*10^6)