Consider a 5x3 rectangle- it is 5 unit squares wide and 3 unit squares tall, thus made up of 15 little squares. If you shade the outside squares, the border squares, 12 squares are shaded, with 3 in the middle left unshaded.

Can you make rectangle with its border squares shaded so that the number of shaded squares equals the number of squares in the center? If so, how many rectangles (including squares) like that can you make? What are the dimensions of those rectangles?

2.

A guy was having a paper written in English translated into French. He got the assistance of a French translator named Jacques. At the bottom of the paper was the following: (written in French of course)

"Much thanks to my friend Jacques for translating the above paper into French."

and then:

"More thanks to my friend Jacques for translating the above sentence into French."

and then:

"More thanks to my friend Jacques for translating the above sentence into French."

At first glance, this would have to continue forever, for proper thanks to be due. But it doesn't need too... it can (and does) end right there. Why?