It is a common joke among Finnish players, that one's dan rating is a direct measure of his/her counting ability. That is, a shodan can count to one, 2-dan can count to two and so on.
Try it: listen to a dan player while his groups are involved in a complicated battle. If he goes "one-two-three-*duh*, one-two-three-*duh", you can be quite sure he is a 3-dan.
Of course, this would mean that kyu players cannot count at all, which is not too far from the truth. Extending from the above principle, you possibly could estimate a kyu rank by measuring how many mistakes happen between correct moves. This will sound like "*duh*-*duh*-*duh*-*duh*-one" for a 4 kyu.
Since this page is called Dan Mathematics, here's another:
We once had four player next to a computer, and were wondering what strength should we report to the go server.
The individual players' strengths were 3-dan, 1-kyu, 1-kyu and 2-kyu. So, just adding them up (3-dan being interpreted as -2 kyu, which produces the correct handicap) we got -2-kyu + 1-kyu + 1 kyu + 2kyu = 2 kyu.
The fun part is, when evaluating this same sum in "dan mathematics", the result is 3-dan + 0-dan + 0-dan + -1-dan = 2-dan.
So, clearly we were both 2-kyu and 2-dan at the same time..
No wonder the roman numerals caused the collapse of their empire.
Actually, you forgot to average it. Since it was four people, the result would be .5 dan and .5 kyu, or exactly halfway between one kyu and one dan.
I thought if you were white you get an extra 5.5 (or whatever it is these days) komi. ~srn347