1 ! Copyright (C) 2009 Jon Harper.
2 ! See http://factorcode.org/license.txt for BSD license.
3 USING: project-euler.common math kernel sequences math.functions math.ranges prettyprint io threads math.parser locals arrays namespaces ;
6 ! http://projecteuler.net/index.php?section=problems&id=255
10 ! We define the rounded-square-root of a positive integer n as the square root of n rounded to the nearest integer.
12 ! The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of n:
14 ! Let d be the number of digits of the number n.
15 ! If d is odd, set x_(0) = 2×10^((d-1)⁄2).
16 ! If d is even, set x_(0) = 7×10^((d-2)⁄2).
19 ! until x_(k+1) = x_(k).
21 ! As an example, let us find the rounded-square-root of n = 4321.
22 ! n has 4 digits, so x_(0) = 7×10^((4-2)⁄2) = 70.
24 ! Since x_(2) = x_(1), we stop here.
25 ! So, after just two iterations, we have found that the rounded-square-root of 4321 is 66 (the actual square root is 65.7343137…).
27 ! The number of iterations required when using this method is surprisingly low.
28 ! 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).
30 ! 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))?
31 ! Give your answer rounded to 10 decimal places.
33 ! Note: The symbols ⌊x⌋ and ⌈x⌉ represent the floor function and ceiling function respectively.
37 : round-to-10-decimals ( a -- b ) 1.0e10 * round 1.0e10 / ;
39 ! same as produce, but outputs the sum instead of the sequence of results
40 : produce-sum ( id pred quot -- sum )
41 [ 0 ] 2dip [ [ dip swap ] curry ] [ [ dip + ] curry ] bi* while ; inline
44 number-length dup even?
45 [ 2 - 2 / 10 swap ^ 7 * ]
46 [ 1 - 2 / 10 swap ^ 2 * ] if ;
47 : ⌈a/b⌉ ( a b -- ⌈a/b⌉ )
50 : xk+1 ( n xk -- xk+1 )
51 [ ⌈a/b⌉ ] keep + 2 /i ;
53 : next-multiple ( a multiple -- next )
54 [ [ 1 - ] dip /i 1 + ] keep * ;
57 ! Gives the number of iterations when xk+1 has the same value for all a<=i<=n
58 :: (iteration#) ( i xi a b -- # )
60 [ drop i b a - 1 + * ]
61 [ i 1 + swap a b iteration# ] if ;
63 ! Gives the number of iterations in the general case by breaking into intervals
64 ! in which xk+1 is the same.
65 :: iteration# ( i xi a b -- # )
70 ! set up the values for the next iteration
71 [ nip [ 1 + ] [ xi + ] bi ] 2keep
72 ! set up the arguments for (iteration#)
73 [ i xi ] 2dip (iteration#)
75 ! deal with the last numbers
76 [ drop b [ i xi ] 2dip (iteration#) ] dip
79 : 10^ ( a -- 10^a ) 10 swap ^ ; inline
81 : (euler255) ( a b -- answer )
83 [ [ drop x0 1 swap ] 2keep iteration# ] 2keep
89 : euler255 ( -- answer )
90 13 14 (euler255) round-to-10-decimals ;