Two friends, Stewart and Ian, sat down to discuss what they had been working on, recently, in preparation for the upcoming puzzling competition in their school. Today's discussion centred on five closely related problems based on a well known set of puzzles but, for which, the boys had added their own slight variation.
"The answer to the first one is obvious, I think," said Stewart.
"Yes," agreed Ian, "even using a simple brute-force approach you can't help but stumble on the right answer."
"A brute force approach is also useful here, provided you proceed this way," said Stewart, indicating his strategy.
Ian scribbled down some numbers on his sheet of paper and, showing it to Stewart, declared, "My way of thinking of it was to represent the problem like so and only use those with two odd numbers."
"This one I found a bit trickier," admitted Stewart, "but I think the answer might be this which I deduced using a strategy I call 'topless left, topless right'."
"In fact, I can prove your answer is correct," declared Ian, rather smugly. "We can break our set up into equivalence classes like so," he said, as he wrote. "I can choose one member from each class but not simultaneously from both this one and this one."
"I thought this one would be much more difficult," said Stewart, "but then I realised the solution was staring me in the face".
"Egad, you're right!" exclaimed Ian.
"I have to be honest, I couldn't figure out this last one," said Stewart. "From the first problem, I know the answer must be less than or equal to $10$ and some rudimentary work has shown me it is at least $7$."
"I have thought a lot about this one but haven't finished it." said Ian. "My hypothesis is that the answer is $9$ and I stayed up all night trying to prove it."
Stewart proceeded to take out his laptop and said, "Let's ask our old friend, the internet, shall we?" After a few minutes of searching, Stewart closed the laptop again, looked sympathetically at Ian and said, "I'm afraid it's back to the drawing board on this one, buddy."
What are the problems the boys are trying to solve?
What are the answers in each case?
"For Problem 4, were you able to prove your answer?" asked Ian.
"Why yes," replied Stewart, "you can divide the whole set into duelling pairs and ..."
"Ah, I see," said Ian, "only one allowed in each pair."
"There seem to be many different configurations corresponding to the answer to Problem 1," remarked Stuart.
"That's true," said Ian, "more than three and half million. But I just chose a diagonal approach."