-! Copyright (C) 2009 Jon Harper.
+! Copyright (c) 2009 Jon Harper.
! See http://factorcode.org/license.txt for BSD license.
-USING: project-euler.common math kernel sequences math.functions math.ranges prettyprint io threads math.parser locals arrays namespaces ;
+USING: arrays io kernel locals math math.functions math.parser math.ranges
+ namespaces prettyprint project-euler.common sequences threads ;
IN: project-euler.255
! http://projecteuler.net/index.php?section=problems&id=255
! DESCRIPTION
! -----------
-! We define the rounded-square-root of a positive integer n as the square root of n rounded to the nearest integer.
-!
-! The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of n:
-!
-! Let d be the number of digits of the number n.
-! If d is odd, set x_(0) = 2×10^((d-1)⁄2).
-! If d is even, set x_(0) = 7×10^((d-2)⁄2).
-! Repeat:
-!
-! until x_(k+1) = x_(k).
-!
+
+! We define the rounded-square-root of a positive integer n as the square root
+! of n rounded to the nearest integer.
+
+! The following procedure (essentially Heron's method adapted to integer
+! arithmetic) finds the rounded-square-root of n:
+
+! Let d be the number of digits of the number n.
+! If d is odd, set x_(0) = 2×10^((d-1)⁄2).
+! If d is even, set x_(0) = 7×10^((d-2)⁄2).
+
+! Repeat: [see URL for figure ]
+
+! until x_(k+1) = x_(k).
+
! As an example, let us find the rounded-square-root of n = 4321.
! n has 4 digits, so x_(0) = 7×10^((4-2)⁄2) = 70.
-!
+
+! [ see URL for figure ]
+
! Since x_(2) = x_(1), we stop here.
-! So, after just two iterations, we have found that the rounded-square-root of 4321 is 66 (the actual square root is 65.7343137…).
-!
+
+! So, after just two iterations, we have found that the rounded-square-root of
+! 4321 is 66 (the actual square root is 65.7343137…).
+
! The number of iterations required when using this method is surprisingly low.
-! For example, we can find the rounded-square-root of a 5-digit integer (10,000 ≤ n ≤ 99,999) with an average of 3.2102888889 iterations (the average value was rounded to 10 decimal places).
-!
-! Using the procedure described above, what is the average number of iterations required to find the rounded-square-root of a 14-digit number (10^(13) ≤ n < 10^(14))?
-! Give your answer rounded to 10 decimal places.
-!
-! Note: The symbols ⌊x⌋ and ⌈x⌉ represent the floor function and ceiling function respectively.
-!
-<PRIVATE
+! For example, we can find the rounded-square-root of a 5-digit integer
+! (10,000 ≤ n ≤ 99,999) with an average of 3.2102888889 iterations (the average
+! value was rounded to 10 decimal places).
+
+! Using the procedure described above, what is the average number of iterations
+! required to find the rounded-square-root of a 14-digit number
+! (10^(13) ≤ n < 10^(14))? Give your answer rounded to 10 decimal places.
+
+! Note: The symbols ⌊x⌋ and ⌈x⌉ represent the floor function and ceiling
+! function respectively.
-: round-to-10-decimals ( a -- b ) 1.0e10 * round 1.0e10 / ;
+! SOLUTION
+! --------
+
+<PRIVATE
! same as produce, but outputs the sum instead of the sequence of results
: produce-sum ( id pred quot -- sum )
[ 0 ] 2dip [ [ dip swap ] curry ] [ [ dip + ] curry ] bi* while ; inline
: x0 ( i -- x0 )
- number-length dup even?
+ number-length dup even?
[ 2 - 2 / 10 swap ^ 7 * ]
[ 1 - 2 / 10 swap ^ 2 * ] if ;
+
: ⌈a/b⌉ ( a b -- ⌈a/b⌉ )
[ 1 - + ] keep /i ;
DEFER: iteration#
! Gives the number of iterations when xk+1 has the same value for all a<=i<=n
:: (iteration#) ( i xi a b -- # )
- a xi xk+1 dup xi =
- [ drop i b a - 1 + * ]
- [ i 1 + swap a b iteration# ] if ;
+ a xi xk+1 dup xi =
+ [ drop i b a - 1 + * ]
+ [ i 1 + swap a b iteration# ] if ;
! Gives the number of iterations in the general case by breaking into intervals
! in which xk+1 is the same.
:: iteration# ( i xi a b -- # )
- a
- a xi next-multiple
- [ dup b < ]
- [
+ a
+ a xi next-multiple
+ [ dup b < ]
+ [
! set up the values for the next iteration
[ nip [ 1 + ] [ xi + ] bi ] 2keep
! set up the arguments for (iteration#)
- [ i xi ] 2dip (iteration#)
- ] produce-sum
+ [ i xi ] 2dip (iteration#)
+ ] produce-sum
! deal with the last numbers
[ drop b [ i xi ] 2dip (iteration#) ] dip
+ ;
-: 10^ ( a -- 10^a ) 10 swap ^ ; inline
-
-: (euler255) ( a b -- answer )
+: (euler255) ( a b -- answer )
[ 10^ ] bi@ 1 -
[ [ drop x0 1 swap ] 2keep iteration# ] 2keep
swap - 1 + /f ;
-
PRIVATE>
-: euler255 ( -- answer )
- 13 14 (euler255) round-to-10-decimals ;
+: euler255 ( -- answer )
+ 13 14 (euler255) 10 nth-place ;
-SOLUTION: euler255
+! [ euler255 ] gc time
+! Running time: 37.468911341 seconds
+SOLUTION: euler255