# The ghost of the sea

Suppose you are given an infinite amount of missiles. Each second, you can shoot at one spot on the integer axis. Is it possible to hit the submarine in finite time?

*Source: Brilliant.*

**Yes.**

Let \(x'\) be the starting point of the submarine, let \(v'\) be its velocity. Thus, the submarine has a certain pair of unknown values \((x',v')\in\mathbb{Z}\times\mathbb{Z}\). As \(\mathbb{Z}\) is a countable set, \(\mathbb{Z}\times\mathbb{Z}\) is countable as well, so we can find a bijection \(f\) from \(\mathbb{N}^\times\) to \(\mathbb{Z}\times\mathbb{Z}\). If we shoot at the number \(x_t+v_tt\) on the \(t^\text{th}\) second, with \(f(t)=(x_t,v_t)\), then, eventually, we will reach the exact location of the submarine. Convince yourselve that this will happen within at most \(f^{-1}\big((x',v')\big)\) seconds.

Note that, as any direct product of countable sets is countable, one can easily generalise this problem by adding acceleration, jerk, snap, crackle, pop..., as long as these variables take discrete (countable) values.

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