Questions from economics honors exam at Oberlin College

Steven Landsburg was chosen by the economics department at Oberlin College to be an outside examiner to "determine who among its top graduating seniors should receive an honors degree." He posted the written exam, which consists of 10 questions, to his blog.

I feel confident in stating that if I took the test I would get a score of 0.

Question 6. When Eve works, she produces exactly one apple per hour. Adam is completely unproductive and can produce nothing at all. Eve's income is taxed at a flat percentage rate, with the proceeds delivered to Adam. What determines the optimal tax rate? What does "optimal" mean here, and what philosophical justification would many economists give for adopting this tax rate?

To make the problem concrete, you can assume that both Adam and Eve, if it were both possible and necessary, would be willing to work up to 1 hour for 1 apple, up to 2 hours for 4 apples, up to 3 hours for 9 apples, and up to x hours for x^2 apples. Now what is the optimal tax rate? (Your answer should be a number.)

Question 8. The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he takes the king's place, becomes the new king, and awaits the next Duke's arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke's first priority is to remain alive, and his second priority is to become king. Who is king on May 6?