Spot is a dog. Every dog has one alpha (also a dog), and no two dogs have the same alpha. But Spot, alone amongst dogs, isn’t anybody’s alpha. 🙁

For any two dogs, there is an assigned referee in case they get into a fight and an assigned marriage counselor in case they get married. The dogs have set up the following rules for deciding who will be the referees and counselors for who:

If Spot fights with any dog, the other dog gets to be the referee. The referee of a fight between dog 1’s alpha and dog 2 has to be the alpha of the referee of a fight between dogs 1 and 2.

Spot has to be his own marriage counselor, no matter who he marries. The marriage counselor for dog 1’s alpha and dog 2 has to referee any fight between dog 2 and the marriage counselor for dog 1 and dog 2.

Finally, dog 1 is stronger then dog 2 if and only if dog 1 is the referee for dog 2 and some other dog. Strength is transitive, and no dog is stronger than itself.

Question 1: Who’s the marriage counselor for Spot’s alpha and the referee of a fight between Spot’s alpha and Spot’s alpha?

Question 2: How many dogs are there?

Question 3: Is the referee for dog 1’s alpha and dog 2 always the same as the referee for dog 2’s alpha and dog 1? What if we asked the same question about marriage counselors?

Question 4: Is any dog stronger than their own alpha?

Bonus Question: Is it possible for there to be a dog that’s stronger than Spot, Spot’s alpha, Spot’s alpha’s alpha, and so on?

A challenge: Try to figure out a trick that allows you to figure out the above questions in your head. I promise, it’s possible!