When Chelsea posted the link to the entirety of Thom McKee's run, I decided to embark on a little project: How random was the Tic Tac Dough bonus round?
As it turns out, not very random (but random enough that it probably wasn't worth memorizing patterns in order to win moderately desirable merchandise). Whether this was the result of the computer being programmed with predetermined patterns or simply the fact that the "random" function in early computer languages wasn't particularly random, I don't know.
During the 46 episodes that Thom appeared in, a contestant faced the dragon 45 times. Across all those bonus rounds, there were only 14 different configurations that were seen, ranging in frequency during that 45–bonus round span from one to seven times. Once in a while, the same configuration would show up twice in a row (B-175 and B-177; C-011 and C-012; C-013 and C-014).
One interesting anomaly is that during the 45 bonus rounds, the dragon never appeared in box 7. Jenny accidentally discovered this when she began making a habit of always offering 7 as her first pick. We saw Thom find the dragon on his first pick a couple of times during his run, so we know it was possible to immediately hit the dragon -- it just plain wasn't in box 7.
Of the 14 different configurations seen, the dragon was in...
...box 1: 2 times
...box 2: 1 time
...box 3: 2 times
...box 4: 1 time
...box 5: 1 time
...box 6: 3 times
...box 7: 0 times
...box 8: 2 times
...box 9: 2 times
Of the 43 times Thom McKee faced the dragon, the dragon was in...
...box 1: 7 times
...box 2: 4 times
...box 3: 5 times
...box 4: 3 times
...box 5: 5 times
...box 6: 8 times
...box 7: 0 times
...box 8: 5 times
...box 9: 6 times
The category shuffling in the main game was also subject to technology limitations, but because of the effort involved in tracking those patterns (especially once Xs and Os started showing up), I didn't bother. I did note that in episodes B-168, B-172, B-184, B-185, and B-192, the very first shuffle of a round left all nine categories in the exact same place they started.