All Roads Lead to Rome – 2
This map shows a collection of one-way roads going between cities A, B, C, and D. To get from A to D you can either go through B or C. There are 3 x 2 = 6 ways of going from A to D if you go through B. There are 2 x 4 = 8 ways of going from A to D if you go through C. Therefore, there are a total of 6 + 8 = 14 ways of going from A to D.
THE CHALLENGE
Here is a complicated pretend map of one-way roads leading from Paris to Rome. How many possible routes are there?
EXPLORATION
Why does this puzzle not work using two-way roads? Make some fun maps for your friends to puzzle over.
Notes
THE CHALLENGE
This puzzle involves several important counting methods.
- If you have independent ways to do two steps of something, the total number of ways is the product of the number of ways of doing each of the steps..
- If two things are completely separate, then their ways add up.
- You can sometimes count things by flooding out from a beginning point.
Let’s start in Paris and make our way to Rome.
- E – There are 3 ways to get to E from Paris.
- H – There are 2 ways to get here directly from Paris, and 3 x 1 = 3 ways of getting to H from E. Therefore, there are a total of 2 + 3 = 5 ways of getting to H from Paris.
- F – 3 ways to go from Paris to E; 2 ways to get from E to F; so 3 x 2 = 6 ways from Paris to F.
- G – There are two kinds of routes for getting to G – either directly from E or from F. There are 3 x 2 ways of getting to G straight from E. There are 6 x 1 = 6 ways of getting to G from F. Thus, there are a total of 6 + 6 = 12 ways of getting to G.
- K – You can get to K either from H or E. Going through H there are 5 x 3 = 15 ways. Going through E there are 3 x 1 = 3 ways. Altogether there are 15 + 3 = 18 ways of getting to K.
- Rome – The last step of going to Rome can be through F, G, or K. Through F there are 6 x 2 = 12 ways. Through G there are 12 x 1 = 12 ways. Through K there are 18 x 3 = 54 ways. That makes a grand total of 12 + 12 + 54 = 78 ways of going from Paris to Rome.
EXPLORATION
If you have two-way roads, that would create loops that would create an infinite number of possible routes.