Thanos and Particles Riddles

Here are two riddles, the solutions of which I will discuss next week. Interested readers are welcome to email me solutions.

Thanos Problem

Thanos, the all-powerful supervillain, can snap his fingers and destroy half of all the beings in the universe.

Suppose now there are $n$ Thanoses, each snapping his fingers in a sequence, one after the other. When a Thanos snaps his finger, each being (Thanos or human) dies independently with probability $1/2$ .

Question: Out of $K$ people on Earth, how many can we expect to still be alive at the end? More formally, let $p_n$ be the probability that a human being survives. Initial values: $p_1=1/2$ , $p_2=3/8$ .

Show that $p_n = \Theta(1/n)$ , that is, $n \cdot p_n$ is bounded between positive constants.
Does $(p_1 + \ldots + p_n)/ \log{n}$ converge? If so, to what limit?
Does $n \cdot p_n$ converge? If so, to what limit?

Note: this problem was posted (with fewer details) by Oliver Roederon on the 538 blog, but without an analytical solution. The plot is due to Laurent Lessard (github).

Particles Problem

Suppose there are $n$ particles in the unit square. Initially one particle is awake and all others are sleeping. Each awake particle moves in the unit square at speed $1$ in a direction you prescribe and wakes up any sleeping particle it encounters. The particles that are awake move simultaneously and particles can change direction at any point in time.