Nice Challenge

Basically you need to check for any string which begins and ends with the same character, any sequence of 0s or 1s inside this will match the '10|01' criteria. You also need to account for '1', '0', '11' and '00' as these also meet the '01|10' constraint.

/

^(0|1)$       # simple cases of 1 or 0,  with zero 01|10
   |
^1(0|1)*1$    # 1 .. anything .. 1
   |
^0(0|1)*0$    # 0 .. anything .. 0

/x
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my @tests = qw(
           0 1 
           01 10 00 11 
           000 111 001 011 101 010 100 110 
           1001 1101 0101 1010
          );

foreach (@tests) {
  print "$_ ";
  if (/^(0|1)$|^1(0|1)*1$|^0(0|1)*0$/) {
    print "passed\n";
  }
  else {
    print "failed\n";
  }
}
[download]