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A nation has A airports with 1 in each city. Each city has direct flights to 3 other cities, and each of them has a return flight. Also, people can fly from a city to another with 1 layover at most. What is the maximum value of A?

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    In graph theory terms, it sounds like you're asking for the largest number of vertices in a 3-regular graph with diameter 2. Is that right? – 2017-02-25
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    This is known as a [Moore Graph](https://en.wikipedia.org/wiki/Moore_graph) and you can find some info on it under that name. – 2017-02-25
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    @GTonyJacobs I had forgotten the diameter terminology ... but YES , that is my understanding of the problem ... Is the maximal solution going to be the Peterson graph ? – 2017-02-25
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    Thanks for the answer... which is now gone! I was able to connect all 10 nodes together! I am not entirely sure what the Moore graph is, but I'll certainly look into it. – 2017-02-25

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Consider one city, Startville. From here there are three flights to cities Alpha, Bravo and Charlie, and from those three cities there are two more cities that could be connected (because there is a flight back to Startville also for those cities). So the limit is definitely no more than $1+3+6=10$ cities, but is that many possible?

Alpha, Bravo and Charlie are already connected under the one-layover rule, via Startville, but what about the cities beyond them? We can easily arrange for $3$ such cities, Foxtrot, Golf and Hotel, just connected in a triangle, which leaves a "spare" connection at Alpha, Bravo, Charlie for a fourth city Xanadu. So I can get $8$:

enter image description here

BUT

I don't promise that it is the maximum. $8$ is demonstrated, $10$ is an upper limit.

Edit to add: And of course @DonaldSputterwit reminds me in the comments that if you want a small cubic graph with unusual properties the Petersen graph is always worth looking at:

enter image description here

which does the trick for the maximum $10$ cities.

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    Do not connect F,G & H into a triangle ... my money is on the maximal value 10 ... The Peterson graph. – 2017-02-25
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Here's a solution with 10 nodes:

solution with 10 nodes

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    @celtchk 10/10 for the solution ... 11/10 for the drawing .. lol – 2017-02-26