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Euler problems/101 to 110

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< Euler problems
Revision as of 17:19, 30 November 2007 by MartinL (Talk | contribs)

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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:

problem_108 = undefined

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:

problem_110 = undefined