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 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 .

* Question:* Out of people on Earth, how many can we expect to still be alive at the end? More formally, let be the probability that a human being survives. Initial values: , .

Show that , that is, is bounded between positive constants.

Does converge? If so, to what limit?

Does 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 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 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.

* Question:* Show that you can wake up all the particles by time .

*Note: I learned this problem from Maria Gringlaz.*

*New: A solution to the particles problem is now available here.*

## 2 thoughts on “Thanos and Particles Riddles”