how many hexagons can be found in the graph?

Question

At first sight I thought that this question requires considering different cases, but I find it difficult to convince myself as to how to start this question. Could someone please give me some hints?

Answer 1

The ways to make a hexagon from squares in this configuration are:

 _____   ________   ________
|__|__| |__|__|__| |__|__|__|
|__|    |__|       |__|__|

As well as the rotations of each of these. This means there are $4$ of each of the 4-hexagons and 5-hexagons.

For the 3-hexagons, we know there must be at least six (placing the corner in each of the six original squares), but the middle two squares could have the horizontal part going left OR going right, so they each have two possibilities. This brings us to $8$ of the 3-hexagons.

So the total is $16$.

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EDIT:

To make it a little clearer, consider the ways you can pick one, two, three, four, or five of the squares so that they are connected along the edges:

 __   _____   _____   ________   ________
|__| |__|__| |__|__| |__|__|__| |__|__|__|
             |__|    |__|       |__|__|

 ________   ________   _____   ________
|__|__|__| |__|__|__| |__|__| |__|__|__|
           |__|  |__| |__|__|    |__|

 _____
|__|__|__
   |__|__|

And any reflections/rotations of these. Notice that only the ones from the first set (from the original answer) are six-sided. This means that we don't need to worry with the remaining figures.

So we can identify all the hexagons just by placing the three from the top in all the various spots:

+-- ++- ++- -+- -+- -++ -++ --+
++- +-- -+- ++- -++ -+- --+ -++

--+ +-- +++ +++ +++ +++ -++ ++-
+++ +++ +-- --+ ++- -++ +++ +++

In this diagram, pluses are squares you want, and minuses are squares you don't.