Why use a special variable in an unintended and undocumented manner when you can use ($t^=1)?
Change
local $| = 0; $sum += $chars[$_] * (--$| ? 3 : 1) foreach 0..10;
to:
my $t = 0; $sum += $chars[$_] * (($t^=1) ? 3 : 1) foreach 0..10;
Update: sgifford improved this to:
my $t = 1; $sum += $chars[$_] * ($t^=2) foreach 0..10;
Update:
my $check = (10 - $sum % 10) % 10;
simplifies to
my $check = -$sum % 10;
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