[Kc] 100 Monkeys: Solved

[Garrett Goebel]

100 Monkeys: 2003-01-14 

[- Problem -]
There are 100 doors, all closed. In a nearby cage are 100 monkeys. The first
monkey is let out, and runs along the doors opening every one. The second
monkey is then let out, and runs along the doors closing the 2nd, 4th,
6th,... all the even-numbered doors. The third monkey is let out. He attends
only to the 3rd, 6th, 9th,... doors (every third door, in other words),
closing any that is open and opening any that is closed. The fourth monkey
does the same for the 4th, 8th, 12th, 16th,... doors, opening the closed
ones and closing the open ones. The fifth monkey does the same to the 5th,
10th, 15th,... doors, and so on. After all 100 monkeys have done their work
in this way, which doors are left open?


[- Answers -]

__PAUL_KULCHENKO__
for$a(1..(@c=(0)x100)){my$b;$c[($b+=$a)-1]^=1 for 1.. at c/$a}

__GENE_DASCHER__
my %doors;
my $OPEN=1;
my $CLOSED=0;
my $door_string = "";

for (my $x = 1; $x <= 100; $x++) {
  my $y = $x;
  while ($y <= 100) {
    $doors{$y} = ((defined $doors{$y})
      ? (($doors{$y} == $OPEN) ? $CLOSED : $OPEN)
      : $OPEN);
    $y += $x;
  }
}

foreach $door (sort {$a <=> $b} keys %doors) {
  $door_string .= ((($doors{$door} eq $OPEN))
    ? ((($door_string eq "") ? "" : " ") . $door) : "" );
}


[- Discussion -]
I found out the probable source of the puzzle:
http://olimu.com/Notes/Monkeys&Doors.htm

Feel free to post questions about why and how the following code does what
it is supposed to do. 

I for one was surprised by @c=(0)x100. I thought x 100 could only be used on
strings. For instance to get string of 100 0's is

  $c='0'x100

I had no idea you could use the syntax Paul provided to initialize an array
with 100 scalar zero elements. Add on to that, he wraps the assignment in
parens and immediately uses @c in a scalar context which returns the number
of elements in the array... which is used to specify the range of the
foreach style for loop.

  for$a(1..(@c=(0)x100))

The parens avoid an error which would otherwise occur because perl would
read something like 1.. at c=1 as attempting to modify a range in scalar
assignment. So what we wind up with is extremely compact notation for the
construction a loop which increments $a from 1 to 100. I'm not sure where
I'd necessarily need to use this, but I like it ;)

I was also surprised that Paul's answer while both solving the problem and
matching the regex I'd specified, was in a totally different format than I'd
imagined. Just goes to show what happens when you don't nail down your specs
;)


[- Algorithmic Solution -]

All the submissions solved the problem with brute force. I was a bit
disappointed that nobody submitted an answer which solved it
algorithmically. So I've included one myself below. The trick with 100
monkeys, is to look at the pattern that emerges, recognize that they are
perfect squares and figure out why.

In case you're still wondering, the doors left open are: 1 4 9 16 25 36 49
64 81 100

These numbers as you can see are perfect squares. Why? Each door is visited
by a limited set of monkeys. The 9th door will be visited by monkeys 1, 3,
and 9. Door 10 by monkeys 1, 2, 5, and 10. If you look at that long enough,
you'll realize that the monkeys that will visit the door are the monkeys
who's number can exactly divide the door's number... the factors. And it
just so happens that perfect squares always have an odd number of distinct
factors. So it follows that those doors which are numbered with perfect
squares will only be toggled an odd number of times. Consequently you can
skip the brute force solution and solve it algorithmically with:

  join' ',map$_**2,1..10


--
Garrett Goebel
IS Development Specialist

ScriptPro                   Direct: 913.403.5261
5828 Reeds Road               Main: 913.384.1008
Mission, KS 66202              Fax: 913.384.2180
www.scriptpro.com          garrett at scriptpro dot com
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