tool changer logic- mathematical challenge

Test yourself, see how good you really are...

Given; a ratcheting action, rotary tool turret with 8 positions (a
tool every 45 degrees). The turret can only index going one direction.

Given; a series of 4 optical sensors, spaced evenly @ 90 degrees
around a circle.

Given; a "C" shaped, semi-circular raised area that passes thru and
blocks/unblocks the optical sensors as it rotates with the tool
turret. This creates 2 state change areas for each of the 4
sensors. Assuming the "C" shape, and CCW motion, you can see that
each of the 4 sensors will go from blocked to unblocked, and
unblocked to blocked as the turret rotates.

Problem; how many degrees is the opening of the semi-circular "C"
shape that will create an off - on and on - off transition for each
of the 4 sensors. Resulting in an index resolution of 45 degrees
when only looking at one sensor at a time.


This is easier than it first appears, however, I never could figure
it out without help. I thought this might be an interesting
challenge for others. See what you come up with. Good luck.