The numbers on a circle

Knowing that cos^2 (theta) + sin^2(theta) = 1, let 1 corresponds to theta = 0, 2 correspond to theta = pi/30. Then, 7 is 6*pi/15, 16 is pi, 30 is 29*pi/15. I took a very pedantic interpretation of "the numbers are evenly spaced"- that is, 1 and 30 do not both sit on the coordinate (1, 0).

Then, what is diametrically opposite can be recovered by 6*pi/15 + pi = 21*pi/15. This is 22.

I stared at it, and I realized that working directly in Cartesian coordinates would be inconvenient; thus, I turned to that trigonometry identity and polar coordinates, where I can actually spell out the coordinates in a convenient way.

Some extensions based off of this puzzle:

• If you were to draw another diametric line such that it is 90 degrees with the one drawn from 7 to 22, which two numbers would it be?
• Choose any n you want where 1 < n < 15. Between which pairs of numbers would you draw the diametric lines if you want them all to have the same angle between them?
• Impossible puzzle: what if, rather than from 1 to 30, it was from 1 to 31? Suppose that 1 is theta = 0, 2 is theta = pi/15.5, 3 is 2*pi/15.5, ..., 31 is 30*pi/15.5. Now, 7 is 6*pi/15.5, so the "opposite" side would be 21.5*pi/15.5... what is this, halfway between 22 and 23?

I suppose what makes a puzzle geometrical rather than logical is the presence of measurement of space and shape. LSAT questions, for instance, emphasize decoding information precisely off of a deliberately convoluted text- something that requires some good logic chops, but not necessarily spatial reasoning per se.