Experiments you can run yourself.
You can walk every bridge exactly once when the number of vertices of odd degree is 0 or 2.
Any map in the plane can be colored using at most four colors.
As we add cities, the number of routes to check grows explosively (factorially).
A Hamiltonian cycle requires visiting every vertex once (an Eulerian path — every edge).
An efficient algorithm sorts n elements in time of order n·log n, not n².
Always taking the most valuable item (greedily) does not always give the optimum.
On average, your friends have more friends than you do.
You cannot write a program that decides, for every program, whether it will halt.
In any group of 6 people there is always a trio of mutual acquaintances or of mutual strangers.
Each of the cube's 43 quintillion arrangements can be solved in at most 20 moves.