A301337 Number of steps required in the worst case for two knights to find the princess in a castle with n rooms arranged in a line (Castle and princess puzzle).
1, 1, 2, 2, 2, 3, 4, 4, 6, 6, 6, 7, 8, 8, 10, 10, 10, 11, 12, 12
Offset: 1
Examples
For n = 1, there is only room to search, so a(1) = 1. For n = 2, the knights search both rooms, so a(2) = 1. For n = 3, the knights can search the first two rooms twice, so a(3) = 2. For n = 4 and 5, the knights can search the second and the fourth rooms twice, so a(4) = 2 and a(5) = 2.
Links
- Dmitry Kamenetsky, Optimal solutions
Formula
It seems that for n >= 3:
if n = 0 mod 6, then a(n) = (n/6)*4 - 1,
if n = 1 or 2 mod 6, then a(n) = floor(n/6)*4,
if n = 3, 4 or 5 mod 6, then a(n) = floor(n/6)*4 + 2.
i.e., a(n) = A302404(n-2).
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