Euler problems/101 to 110

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Problem 101

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


problem_101 = undefined

Problem 102

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


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Problem 103

Investigating sets with a special subset sum property.


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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.

Problem 105

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


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Problem 106

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


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Problem 107

Determining the most efficient way to connect the network.


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Problem 108

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


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Problem 109

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


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Problem 110

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


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