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Re^2: Puzzle Time

by Athanasius (Monsignor)
on Dec 23, 2012 at 10:21 UTC ( #1010086=note: print w/ replies, xml ) Need Help??


in reply to Re: Puzzle Time
in thread Puzzle Time

++LanX for an excellent answer (way faster than my brute force approach).

Allow me one nit-pick:

9! = 362880 possible combinations are checked in a recursive function.

Throwing in a counter shows that sub append_digit is actually called 986,410 times. Took me a while to work out why...

9! is the number of combinations if every number has exactly 9 digits. But we also allow numbers with 1, 2, ..., 8 digits. So the total number of combinations is:

my $p = 9 + # 1 digit (9 * 8) + # 2 digits (9 * 8 * 7) + # 3 digits (9 * 8 * 7 * 6) + # 4 digits (9 * 8 * 7 * 6 * 5) + # 5 digits (9 * 8 * 7 * 6 * 5 * 4) + # 6 digits (9 * 8 * 7 * 6 * 5 * 4 * 3) + # 7 digits (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2) + # 8 digits (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1); # 9 digits say $p; # = 986,409

(That 1 extra sub call is the first one, when @number is empty.)

:-)

For anyone else trying to understand how this code works, here is my slightly-reworked version which outputs the results in the order found:

my $calls = 0; my $count = 0; my @digits; my @solutions; append_digit(); print "count: $count\n", join("\n", @solutions), "\n"; print "recursive calls: ", $calls, "\n"; sub append_digit { ++$calls; my $number = join '', @digits; if (@digits && ! grep { $number % $_ } @digits) { print "$count: $number\n" unless ++$count % 100; push @solutions, $number; } for my $digit (1 .. 9) { next if $digit ~~ @digits; push @digits, $digit; append_digit(); pop @digits; } }

Hope that’s helpful!


Athanasius <°(((><contra mundum Iustus alius egestas vitae, eros Piratica,


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Re^3: Puzzle Time
by LanX (Canon) on Dec 23, 2012 at 15:25 UTC
    > Allow me one nit-pick:

    Yeah I noticed this, but was too tired to correct it. =)

    Anyway my guess was wrong (9!+8!+7!+...) you got it right.

    > (way faster than my brute force approach).

    If it's about speed you can limit the $maxlevel, because the longest number can't have more than 7 digits:

    1. Evidently 0 is excluded!

    2. Any number this long includes even ciphers. So 5 is excluded! But any number divisible by 5 and 2 must end with a 0

    3. An number from the remaining 8 digits would include 9 and 3, but the digit sum would be 40, which is impossible.

    Even your approach with a brute force loop could compete when only considering 7 digits, cause you don't have the overhead of 1 million function calls.

    Cheers Rolf

    ) and therefor a 7 digit number excludes 4 to be divisible by 9.

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