I also tried the same construction using the four weighted mediants (4a+c)/(4b+d), (3a+2c)/(3b+2d), (2a+3c)/(2b+3d), and (a+4c)/(b+4d) (removing common factors as they arise). At the 10th level, I find that every fraction in the list has denominator congruent to 1 mod 4, and most fractions of this form with small denominator do occur althought a few don't; specifically, 4/13, 6/13, 7/13, and 9/13 haven't occurred by level 10 even though 1/13, 2/13, 3/13, 5/13, 8/13, 10/13, 11/13, and 12/13 all occurred back in level 4 or earlier. The only other proper fractions with denominator less than 30 and congruent to 1 mod 4 that haven't occurred by level 10 are 4/29 and 25/29. I'll probably post something about these questions to MathOverflow unless someone on this list says they're working on them. Jim Propp