There is an enemy submarine located at an unknown integer somewhere on the number line. The submarine is moving at a constant integer distance per minute (also unknown, and can be positive or negative). Every minute, you can fire a missile at an integer on the number line in an attempt to destroy the submarine.
Is there a strategy to strike the submarine in a finite amount of time?
Assume that you move first and that you can fire missiles at the number line for as long you need to.
Each potential path of the submarine can be described as an arithmetic sequence. Our missile strategy is then equivalent to creating a sequence that intersects every possible arithmetic sequence. IE: For any arithmetic sequence a(t) and our missile sequence m(t) there is a value of t where a(t)=m(t).
Let s be the initial term of an arithmetic sequence and d be the common difference. Then the set of all arithmetic sequences can be put into a 1:1 correspondence with the lattice points on a plane (s,d). Group the points into subsets based on the metric abs(s)+abs(d). Within each subset order them with the point on the s-axis first and going around counterclockwise. This gives a sequence starting with (0,0), (1,0), (0,1), (-1,0), (0,-1), (2,0), (1,1), (0,2), (-1,1), (-2,0), (-1,-1), (0,-2), (1,-1), (3,0), (2,1), ....
Now expand each ordered pair into its arithmetic sequence, forming a table:
t | (s,d) | sequence
1 | (0,0) | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2 | (1,0) | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3 | (0,1) | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
4 | (-1,0) | -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...
5 | (0,-1) | 0, -1, -2, -3, -4, -5, -6, -7, -8, -9,-10, ...
6 | (2,0) | 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
7 | (1,1) | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
8 | (0,2) | 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
9 | (-1,1) | -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
10 | (-2, 0) | -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, ...
11 | (-1,-1) | -1, -2, -3, -4, -5, -6, -7, -8, -9,-10,-11, ...
etc....
This table, when infinitely extended, maps every possible arithmetic integer sequence in a 1:1 correspondence with the positive integers. Our missile sequence m(t) is the main diagonal sequence 0, 1, 2, -1, -4, 2, 7, 14, 7, -2, -11, ....
A search on the OEIS finds this is sequence
A304587Edited on June 9, 2019, 1:08 pm
Solution
