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All,
I was having a conversation with blokhead in the CB concerning the most efficient solution to this problem. Not having any idea how to determine the big O notation, I ran a few tests to see how well my solution scaled.

The first thing I noticed was the average number of guesses doubled every time I increased the search group by a factor of 10. Then I noticed something really cool - it also corresponded to a power of 2. This allowed me to calculate the average case scenario for any number

```#!/usr/bin/perl
use strict;
use warnings;

my \$num       = \$ARGV[0] || 47;
my \$length    = length \$num;
my \$base      = 10 ** (\$length - 1);
my \$base_ave  = 2  ** (\$length - 1);
my \$next      = 10 ** \$length;
my \$next_ave  = 2  ** \$length;
my \$ave_diff  = \$next_ave - \$base_ave;
my \$tot_diff  = \$next     - \$base;
my \$distance  = \$num - \$base;
my \$guess     = \$base_ave + (\$distance * \$ave_diff) / \$tot_diff;
print "\$guess\n";
[download]```

So for 12_345 you can expect 16.42, and for 123_456_789 you can expect 262.67 As you can see this scales quite well. While this is the average case, the worst case scenario should still scale reasonably well.

Cheers - L~R

In reply to Re: Triangle Numbers Revisited by Limbic~Region
in thread Triangle Numbers Revisited by Limbic~Region

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