Hint and answer to 2.) Estimate the population of New York City, the number of households in the city, the percentage of them (and other organizations) that have pianos, how frequently each piano will be tuned on average, how many pianos the average tuner tunes, and put these together for a rough estimate. (Such problems are called Fermi problems in honor of the Italian physicist Enrico Fermi.) The annual highway toll is approximately 14 times the number of deaths in the 9/11 attacks.
Answer to 3.) Label the hikers with their times. First 1 and 2 go over (2 minutes), and 1 comes back (1 minute). Then 5 and 10 go over (10 minutes) and 2 returns (2 minutes). Finally 1 and 2 go over (2 minutes). The total is 17 minutes.
Answer to 4.) Since the 100-pound sack of of potatoes was 99 percent water, it consisted of 99 pounds of water and 1 pound of pure potato essence. After the evaporation, the sack weighed X pounds and was 98 percent water and 2 percent potato essence. Thus 2 percent of the new weight X is the 1 pound of potato essence. Since .02X = 1, we can solve to get that X = 50 pounds. The answer is that the potatoes now weigh just 50 pounds. This may seem an apolitical problem, but imagine your stockbroker's fixed fee constituting 1 percent of the original worth of your investment, but 2 percent of its present worth. Then the problem is not necessarily small potatoes.
Answer to 5.) You would take one marble from the box labeled "blue and red." Assume it's red. (Analogous reasoning follows if it's blue.) Since the marble is red and it comes from an incorrectly labeled box saying "blue and red," it must be the box with red marbles only. Thus the box labeled "blue" must have either red marbles only or red and blue marbles. It can't be the box with the red marbles only, so it must be the box with blue and red marbles. Finally the box labled "red" must contain the blue marbles.
John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as of the forthcoming (in December) "Irreligion." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.