Some other examples

2 $2^n < n!$ for $n>2$

3 If any square of a $2^n \times 2^n$ chessboard is removed, the remaining squares can be covered by L-triominoes. (That is, by blocks of size equal to three board squares, connected in an ``L'' shape). (This problem is often given where the square removed from the large chessboard is a corner square - but the proof works for any square.)

4 In a certain country, each town is connected to every other town by a (single) one-way road. Prove that there is one town from which you can drive to any other (you may need to stop in intermediate towns to do it).

5 With the same setup of towns and road as the previous problem, can you show that there is a town from which you can drive to any other, stopping in at most one intermediate town?

6 In the same setup as the previous two problems (but with the additional restriction that there are at least three towns) - can you show that, by changing the direction of at most one road, it is possible to get from ANY town to every other?