Shower Head Jet Separation

Question

The jets on a shower head are arranged in circles that are concentric with the rim. The jets are equally spaced out on each circle and there is at least one radius that intersects every circle at a jet.

The angular separation of two jets on a circle is the size of the angle formed by the two radii of the circle that pass through the jets. All angular separations are integers. For example, on a shower head, there are 10 jets on the inner circle. Hence the angular separation of adjacent jets is 360 / 10 = 36 degrees.

Another shower head has four circles with 10, 20, 30 and 45 jets respectively. Explain why no diameter of the shower head passes through 8 jets.

I know that it has to do with same angular separations for all 4 circles and a 180 degree angle doesn't go through 8 jets but how do I go about this?

Also, if a shower head has an inner circle of 12 jets, a middle circle of 18, and an outer circle of 36, how many radii pass through two jets?

I worked out the angular separations for all of the jets, and saw which ones were common to 2 circles, and I got an answer of $0$. Was I correct in my reasoning? There was also another question asking about how many radii passes through 3 jet holes, and I got 10. Was I correct?

P.S. A few of my questions have been deleted because they have been accused of cheating in the Australian Junior Competition however we are allowed to ask for help and/or answers from anyone or anything except for my teacher and classmates

Answer 1

If a diameter of the shower head passed through eight jets, it would have to pass through four antipodal pairs of jets, one pair per ring. It is easy to find such a pair for the rings with 10, 20 and 30 jets, but there is no such pair for the ring with 45 jets: the angular separation between adjacent jets is 8° here, and 8° does not divide 180°. Therefore no diameter passes through eight jets.

For the second question (12/18/36 jets), the number of radii passing through three jets is $\gcd(12,18,36)=6$. The number of radii passing through exactly two jets may be computed as follows:

• $\gcd(12,18)=6$ (radius through jets in inner/middle rings)
• $\gcd(12,36)=12$ (inner/outer rings)
• $\gcd(18,36)=18$ (middle/outer rings)

From each of the above values, the six radii passing through three jets need to be subtracted. The final answer for the number of two-jet radii is $6+12+18-18=18$.