#2             89. THE POPCORN PROBLEM             
“I have this problem,” Laurie confided to Edna. “I ran out of feed for the chooks, so I have been feeding them popcorn for the past few weeks. The trouble starts when I am sending the eggs to market: whenever I place three in a row, there is a large explosion!”

It sounded like a yolk to me, but I saw that he was serious. They turned to the pile of 6 x 6 egg crates.

“I’ll just have to pack as many eggs in each crate as I can,” continued Laurie.

“I think we could send at least 12,” said Edna cheerfully. “Eight in a ring will be all white, and then one in each corner ... .”

“Don’t do it!” I shouted, but it was too late. There was a loud POP! and I ducked for cover to avoid the rain of fowl debris.

How many eggs could Laurie have sent with no three in a row?

Hints and strategies

Hint 1                Hint 2              Solution               Extensions
HINT 1

Just play with this problem to see if you can find a good solution.
HINT 2

Edna mentions the number 12. Perhaps you could show that this is the maximal possible number of eggs?.
SOLUTION

It is clear that the maximal number of eggs is 12 (two per row). The figure below shows that this number can be attained.

              
EXTENSIONS

1. The obvious way of extending this problem is to vary the size of the egg crate, and the number of eggs allowed in a row. Remember the crate does not have to be square.

2. Another idea would be to vary the line constraint. For example, the restriction might be that no four eggs should form the vertices of a square (with edges parallel to the egg crate lines? or any square?).