## Friday, 27 October 2017

### Problem... Cake

#Combinations, #Rotation, #Computational Thinking

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