> It's also possible to allow tile rotation or mirroring. However, this complicates the nature of the edge constraint; with 90-degree rotation, the edge constraints on horizontals must be the same as the edge constraints on verticals. If mirroring (or 180-degree rotation) is allowed, the edge constraints must be internally symmetric. The advantage is that if rotation or mirroring are allowed, the minimum number of tiles needed decreases. (I haven't done the math to explore how far it goes down; somebody should.)<p>Unless I'm mistaken, counting the number of tiles needed for a complete set when allowing rotations and reflections comes down to the classic necklace counting problem [1]. In particular, we have exactly 4 'beads' (i.e. edges in this case) and n colors (conveniently called colors in both theories). (Well, wikipedia's notation actually would call these 'bracelets' unless you didn't include mirroring.) Wolfram Alpha [2] gives an enumeration of these numbers for the first few, namely:<p>2 colors : 6 tiles, 6 w/ mirrors<p>3 colors: 24 tiles, 21 w/ mirrors<p>4 colors: 70 tiles, 55 w/ mirrors<p>5 colors: 165 tiles, 119 w/ mirrors<p>(Disclaimer: I might have messed up the "with mirrors" numbers.)<p>[1] <a href="http://en.wikipedia.org/wiki/Necklace_(combinatorics)" rel="nofollow">http://en.wikipedia.org/wiki/Necklace_(combinatorics)</a>
[2] <a href="http://www.wolframalpha.com/input/?i=necklace+number" rel="nofollow">http://www.wolframalpha.com/input/?i=necklace+number</a>