MathDB

4

Part of the 11th XMO

Problems(1)

BeeHive!!!

Source: 11th XMO #4

2/13/2023
We define a beehive of order nn as follows:
a beehive of order 1 is one hexagon
To construct a beehive of order nn, take a beehive of order nāˆ’1n-1 and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of n=2,3n=2,3
Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.
Let f(n)f(n) be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find f(n)f(n).
combinatorics