How do equal radii lie on concentric circles?

Question

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with using greatest common divisor, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes:

The jets (or holes) on a shower head are arranged in circles that are concentric with the rim. The jets are equally spaced on each circle and there is at least one radius of the shower head 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 the shower head shown, there are 10 jets on the inner circle. Hence the angular separation of adjacent jets on the inner circle is $360°/10 = 36°$.

a) A shower head has three circles of jets: an inner circle with $12$ jets, a middle circle with $18$ jets, and an outer circle with $36$ jets. How many radii of the shower head pass through just two jets?

b) Another shower head has three circles with $20, 30,$ and $45$ jets respectively. Explain why no diameter of the outer circle passes through six jets.

c) A shower head with three circles of jets has a different number of jets in each circle. Exactly two diameters of the shower head pass through six jets. If the total number of jets. If the total number of jets is $100$, how many jets are there in each circle? Find all combinations.

d) Find the maximum number of jets in a shower head with three circles if only one radius of the shower head passes through three jets, no radius passes through just two jets, and each circle has at least two jets.

Answer 1

Hint: In your figure there are $10$ jets in the inner circle, $12$ in the next circle, and $18$ in the outer circle. That means the angular spacings are $36^\circ, 30^\circ, 20^\circ$. They will line up when the total angles agree. For example, the third jet in the middle circle is $2 \cdot 30^\circ$ from start and the fourth in the outer circle is $3 \cdot 20^\circ$ from start. As these agree, they line up. They will repeat at angles of the LCM of the spacings of two circles.

Answer 2

B) other than 0 and 360 there is no common multiple for 20, 30 and 45. If X is a multiple of all 3 and X +180 is also a multiple of all 3 only it will pass thru 6 jets. It is not the case here.

C) 2 180 3 120 5 72 6 60 9 40 15 24 18 20 30 12 45 8 60 6 90 4

A combination which will add up to 100 72+20+8

So 5 degrees, 18 degrees and 45 degrees.

D) Only one of the radii passes thru 3 points. Let's take that as 0 or 360. So we have to find points which not align other than 0 and 360. We have to find 3 angles which have no common multiple other than 360. 72(5 jets), 120 (3 jets) and 180(2 jets). In all 10.

Answer 3

A) The inner, middle, and outer rings have angular separations $10,20,30$ degrees respectively. $10$ divides both $20$ and $30$, so every jet on the inner and middle rings will automatically fall on the same radius as a jet from the outer ring. $LCM(20,30) = 60$, so the number we are looking for is $360/20 + 360/30 - (360/60)*2 = 18+12-12 = 18$

$LCM(20,30) = 60$, and $360/60 = 6$, so there are $6$ points where all 3 fall on the same radius.

This implies the total number of pairs is $18 - 6 = 12$

B) To pass through $6$ jets on a diameter, we would need the angular separations of all three rings to divide $180$. The angular separations are $18, 12, 8$, which clearly do not satisfy the criterion, as $8$ is not a factor of $180$.

C) Here, we need the angular separations of all three rings to be a factor of $90$ and satisfy $18(\frac1{\theta_1} + \frac1{\theta_2}+\frac1{\theta_3}) = 5$

Letting $\theta_j = 90/k_j $ for some factors of $90$, $k_j$, we get

$k_1+k_2+k_3 = 25$

Factors of $90$ are $1,2,3,5,6,9,10,15,18,30,45,90$

Letting $(k_1,k_2,k_3) = (2, 5, 18), (1,6,18), (6,9,10), (1,9,15)$ we see that $(\theta_1, \theta_2, \theta_3) = (45, 18, 5), (90, 15, 5), (15,10,9), (90, 10, 6)$ satisfy the criterion.

So there are $4$ such combinations, where the number of jets are $(8,20,72), (4,24,72), (24,36,40), (4, 36, 60) $.

D) personally, this part seems best fit for a simple python program, but I'll give it my best shot.

The factors of $360$ are $1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360$

Since there are at least $2$ in each circle, we can cross off $360$ as a possibility. We need to find the $3$ smallest numbers from this list which have pairwise LCM of $360$.

You can narrow the list steadily by working your way up from the bottom and noting that if $ab = 360$ and $a$ divides all (or all but one) factor greater than or equal to $b$, $a$ cannot be one of the desired angles. Using this thought, you can rule out $1,2,3,4,5,6$ as possibilities for angular separations. From here, it seems to be more of a brute force method, which I don't have the desire to do. As said before, I recommend writing a program for this part.

However, in the comments it was pointed out that using the prime factorization of $360$ will be a much more efficient way to do it by hand.