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diff --git a/sample/trick2015/ksk_1/remarks.markdown b/sample/trick2015/ksk_1/remarks.markdown new file mode 100644 index 0000000000..b822dc55c8 --- /dev/null +++ b/sample/trick2015/ksk_1/remarks.markdown @@ -0,0 +1,120 @@ +### Remarks + +The program is run with a positive integer as an argument, e.g., +```shell + ruby entry.rb 27 +``` +It has been confirmed to be run on +``` + ruby 1.9.3p385 (2013-02-06 revision 39114) [x86_64-darwin11.4.2] + ruby 2.0.0p481 (2014-05-08 revision 45883) [universal.x86_64-darwin13] + ruby 2.2.3p173 (2015-08-18 revision 51636) [x86_64-linux] +``` + + +### Description + +The program prints a Collatz sequence started with a given number, +that is, it repeatedly outputs numbers obtained by applying the +following Half-Or-Triple-Plus-One (HOTPO) process to the previous +number: + +> If the number is even, divide it by two, otherwise, multiply it by three and add one. + +until the number becomes 1. Collatz conjectured that no matter from +the process starts it always eventually terminates. This is still +an open problem, hence the program may not terminate for some +numbers. It is known that there is no such exception below 2<sup>60</sup>. + + +### Internals + +The source code does not contain either conditional branch or arithmetic operation. +The trick shall be revealed step by step. + +First, the code is obfuscated by using `%`-notations, +`*`(String#join), `%`-formatting, restructuring, and so on. +Here is an equivalent readable program: +```ruby +n = ARGV[0].to_i +begin + # do nothing +end while begin + puts n + n = (/(.)...\1=/ =~ eval('[",,,,,"'+ '",'*n + ' ?=].join#"].join("3x+1?")')) +end +``` +The line +```ruby + n = (/(.)...\1=/ =~ eval('[",,,,,"'+ '",'*n + ' ?=].join#"].join("3x+1?")')) +``` +performs the HOTPO process. +The `eval` expression here returns a string as explained in detail later. +Since *regex*`=~`*str* returns index of first match of *regex* in *str*, +the regular expression `(.)...\1` must match the string +at index `n/2` if `n` is even and +at `3*n+1` if `n` is odd greater than 1. +The match must fail in the case of `n = 1` so that it returns `nil`. + +The key of simulating the even-odd conditional branch on `n` in the +HOTPO process is an `n`-length sequence of the incomplete fragments +`",` where the double-quote `"` changes its role of opening/closing +string literals alternately. If `n` is even, the string in the `eval` +expression is evaluated as +```ruby + => '[",,,,,"'+ '",' + '",' + '",' + ... + '",' + ' ?=].join#...' + => '[",,,,,"",",",...", ?=].join#...' +``` +where the last double-quote `"` is closing hence the code after `#` is +ignored as comments. Note that `"ab""cd"` in Ruby is equivalent to +`"abcd"`. Therefore the `eval` expression is evaluated into +```ruby + ",,,,,...,=" +``` +where the number of commas is `5+n/2`. +As a result, the regular expression `(.)...\1=` matches `,,,,,=` +at the end of string, that is, at index `5+n/2-5 = n/2`. + +If `n` is odd, the string in the `eval` expression is evaluated as +```ruby + => '[",,,,,"'+ '",' + '",' + '",' + '",' + ... + '",' + ' ?=].join#"].join("3x+1?")' + => '[",,,,,"",",",",...,", ?=].join#"].join("3x+1?")' +``` +where the last element in the array is `", ?=].join#"`. Threfore the +`eval` expression is evaluated into +``` + ",,,,,,3x+1?,3x+1?,...,3x+1?, ?=].join#" +``` +where the number of `,3x+1?` is `(n-1)/2`. As a result, the regular +expression `(.)...\1=` matches `?, ?=` at the almost end of string, +that is, at index `5+(n-1)/2*6-1 = 3n+1`, provided that the match +fails in the case of `n = 1` because the symbol `?` occurs only once +in the string. + +One may notice that the string `3x+1` in the code could be other +four-character words. I chose it because the Collatz conjecture is +also called the 3x+1 problem. + + +### Variant + +The Collatz conjecture is equivalently stated as, + +> no matter from the HOTPO process starts, it always eventually + reaches the cycle of 4, 2, and 1 + +instead of termination of the process at 1. This alternative +statement makes the program simpler because we do not have to care the +case of n = 1. It can be obtained by replacing the regular expression +is simply `/=/` and removing a padding `",,,,,"`. The program no +longer terminates, though. + + +### Limination + +The implementation requires to manipulate long strings even for some +small starting numbers. For example, starting from 1,819, the number +will reach up to 1,276,936 which causes SystemStackError on Ruby 1.9.3. +The program works on Ruby 2.0.0 and 2.2.3, though. + + |