55 lines
1.2 KiB
Factor
55 lines
1.2 KiB
Factor
! Copyright (c) 2008 Eric Mertens.
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! See http://factorcode.org/license.txt for BSD license.
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USING: kernel sequences math math.functions math.ranges locals project-euler.common ;
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IN: project-euler.190
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! http://projecteuler.net/index.php?section=problems&id=190
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! DESCRIPTION
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! -----------
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! Let Sm = (x1, x2, ... , xm) be the m-tuple of positive real numbers
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! with x1 + x2 + ... + xm = m for which Pm = x1 * x22 * ... * xmm is
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! maximised.
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! For example, it can be verified that [P10] = 4112 ([ ] is the integer
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! part function).
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! Find Σ[Pm] for 2 ≤ m ≤ 15.
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! SOLUTION
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! --------
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! Pm = x1 * x2^2 * x3^3 * ... * xm^m
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! fm = x1 + x2 + x3 + ... + xm - m = 0
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! Gm === Pm - L * fm
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! dG/dx_i = 0 = i * Pm / xi - L
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! xi = i * Pm / L
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! Sum(i=1 to m) xi = m
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! Sum(i=1 to m) i * Pm / L = m
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! Pm / L * Sum(i=1 to m) i = m
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! Pm / L * m*(m+1)/2 = m
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! Pm / L = 2 / (m+1)
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! xi = i * (2 / (m+1)) = 2*i/(m+1)
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<PRIVATE
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: PI ( seq quot -- n )
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[ * ] compose 1 swap reduce ; inline
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PRIVATE>
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:: P_m ( m -- P_m )
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m [1,b] [| i | 2 i * m 1+ / i ^ ] PI ;
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: euler190 ( -- answer )
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2 15 [a,b] [ P_m truncate ] sigma ;
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! [ euler150 ] 100 ave-time
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! 5 ms ave run time - 1.01 SD (100 trials)
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SOLUTION: euler190
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