xkcd 2529

This repository contains implementations of the "weirdly concrete" math problem presented in xkcd.com/2529. Note that contrary to a strict reading of the text I have interpreted "the longest line" to mean "the line with the most marbles" because that is what the example given in the comic illustrates.

Trapping

The problem describes a random walker on a two-dimensional grid that never revisits any square. Such a walker will inevitably trap itself eventually. On an infinite grid the random walker will step on 71.7 squares on average before it traps itself.

Results

Varying both parameters

For small values of n and k, either decreasing n or increasing k tends to increase the number of marbles on the longest line. Additionally, odd values of n tend to produce lower results than similar even values. It may be surprising that the result can be below 1, but this is because for sufficiently large values of n the walker traps itself before it gets the chance to drop its first marble.

n \ k	1	2	3	4	5	6	7	8	9	10
1	1.00	2.00	2.33	2.70	3.24	3.45	3.75	4.04	4.34	4.56
2	1.00	2.00	2.20	2.60	3.03	3.27	3.50	3.75	3.99	4.20
3	1.00	2.00	2.12	2.29	2.54	2.75	2.94	3.08	3.19	3.29
4	1.00	2.00	2.10	2.32	2.58	2.82	3.00	3.14	3.25	3.37
5	1.00	1.99	2.05	2.15	2.29	2.44	2.59	2.71	2.80	2.87
6	1.00	1.98	2.05	2.18	2.37	2.54	2.69	2.79	2.87	2.93
7	1.00	1.97	2.00	2.06	2.16	2.25	2.35	2.44	2.50	2.55
8	0.997	1.95	1.99	2.09	2.22	2.34	2.45	2.53	2.59	2.63
9	0.995	1.93	1.95	1.99	2.06	2.12	2.19	2.25	2.29	2.33
10	0.990	1.90	1.94	2.01	2.10	2.19	2.26	2.32	2.36	2.39

Large values of k

Any sufficiently large value of k is equivalent to any other, because the walker will have trapped itself before n*k is reached anyway. When the multiple is larger than about 200 it begins to be irrelevant. As a result, it's interesting to see how varying n alone affects the result for any large value of k.

n	intersections
1	11.0
2	7.44
3	4.57
4	4.37
5	3.33
6	3.34
7	2.75
8	2.81
9	2.42
10	2.47
11	2.20
12	2.24
13	2.04
14	2.06
15	1.90
16	1.92
17	1.79
18	1.80
19	1.70
20	1.70
21	1.62
22	1.60
23	1.54
24	1.52
25	1.46
26	1.45
27	1.40
28	1.38
29	1.33
30	1.31
31	1.27
32	1.25
33	1.21
34	1.19
35	1.15
36	1.13
37	1.10
38	1.08
39	1.05
40	1.03
41	0.999
42	0.978
43	0.952
44	0.931
45	0.908
46	0.887
47	0.865
48	0.845
49	0.823
50	0.805

Performance

Each of the implementations averaged the results of a million runs with n=4 and k=1000. They were run on an Intel Core i5-4200U.

• Rust: 5.5 seconds
• Kotlin: 11 seconds
• C: 13 seconds
• C++: 14 seconds
• Go: 15 seconds
• Java: 24 seconds
• Dart: 50 seconds
• JavaScript: 115 seconds
• Python: 137 seconds