Problem... Cake

#Combinations, #Rotation, #Computational Thinking 

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The problem was adapted from the 'Ice Cream Cake' problem in Peter Winkler's 'Mathematical Mind Benders' (p111 with solution on pp115-118), and via Stan Wagon's take on it in Gil Kalai's blog.  It's likely origin was as problem 31.2.8.3 in Olympiad 31 (1968) Moscow Mathematical Olympiads (p90).

SPOILER/SOLUTION:

The values of \(f\left( \theta  \right)\) for \(\left\{ {\theta |\theta  \in \mathbb{N},1 \le \theta  \le 360} \right\}\) can be generated thus:

\[\begin{array}{l}Let\;k = \left\lceil {\frac{{360}}{\theta }} \right\rceil \\\\f\left( \theta  \right) = 2k\;for\;\left\{ {k|k = \frac{{360}}{\theta },1 \le \theta  \le 360} \right\}\\\\f\left( \theta  \right) = 2k\left( {k - 1} \right){\rm{ for }}\left\{ {k|k \ne \frac{{360}}{\theta },1 \le \theta  \le 360} \right\}\end{array}\]
Giving \(f\left( {181^\circ } \right) = 4\) and \(f\left( \theta  \right)\) for all \(\left\{ {\theta |\theta  \in \mathbb{N},1 \le \theta  \le 360} \right\}\):

θ	f(θ)
1	720
2	360
3	240
4	180
5	144
6	120
7	5304
8	90
9	80
10	72
11	2112
12	60
13	1512
14	1300
15	48
16	1012
17	924
18	40
19	684
20	36
21	612
22	544
23	480
24	30
25	420
26-27	364
28-29	312
30	24
31-32	264
33-35	220
36	20
37-39	180
40	18
41-44	144
45	16
46-51	112
52-59	84
60	12
61-71	60
72	10
73-89	40
90	8
91-119	24
120	6
121-179	12
180-359	4
360	2