Three philosophers are taking a nap under a tree. While they’re asleep, a small boy smears their noses with red berries. When they awake, they each begin to laugh, thinking the other two are laughing at each other.
But then one philosopher stops laughing, realizing his nose is red too. How did he come to this conclusion?
Let’s call the philosopher’s A, B and C. A reasoned that B was confident his nose wasn’t red. If B saw A’s nose wasn’t red, he would be surprised that C was laughing, because C would have nothing to laugh at. But B wasn’t surprised, therefore, A correctly reasoned his nose was smeared.
A similar problem can be found in L.A. Graham’s Ingenious Mathematical Problems and Methods with a range of 1 to 9, but the principle remains the same – the numbers with the smallest difference produce the largest product. You start out with the highest two digits, 7 and 6, then attach 5 and 4, putting the smaller of the two digits with the larger number, giving you 74 and 65. The next two highest digits are 3 and 2, giving you 742 and 653. Finally, you add the 1 to the lower number. Page 80 has the details of that solution.
Jeremy was making $10 an hour at his summer job, but he hated the work. He decided to take a 50% pay cut to work at an easier job. He liked his new job at first, then grew bored and found another job paying 50% more than he was currently making. What was his hourly pay at the third job?