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Euler problems/101 to 110

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1 Problem 101

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


problem_101 = undefined

2 Problem 102

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


problem_102 = undefined

3 Problem 103

Investigating sets with a special subset sum property.


problem_103 = undefined

4 Problem 104

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


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
        rec a b = a:rec b (a+b)
fibo_2nk n k = 
        fkm1 = fibo (k-1)
        fkm2 = fibo (k-2)
        fk = fkm1 + fkm2
        fnp1 = fibo (n+1)
        fnp1sq = fnp1^2
        fn = fibo n
        fnsq = fn^2
        fk*fnp1sq + 2*fkm1*fnp1*fn + fkm2*fnsq
fibo x = 
        threshold = 30000
        n = div x 3
        k = n+mod x 3
        if x < threshold 
        then fibos !! x
        else fibo_2nk n k
findCandidates = rec 0 1 0
        m = 10^9
        rec a b n  =
                continue = rec b (mod (a+b) m) (n+1)
                isBackPan a = (sort $ show a) == "123456789"
                if isBackPan a 
                then n:continue
                else continue
search = 
        isFrontPan x = (sort $ take 9 $ show x) == "123456789"
        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 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.


problem_105 = undefined

6 Problem 106

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


problem_106 = undefined

7 Problem 107

Determining the most efficient way to connect the network.


problem_107 = undefined

8 Problem 108

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


import List
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?


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.


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