A group of well-known scientists calling itself Science Debate 2008 (link on page 4) has just called for a presidential debate on scientific issues. Such a forum would be a most-welcome development, but I would supplement it with something more revealing of mental firepower -- garden-variety puzzles.

Big high-tech corporations such as Google and Microsoft as well as a host of smaller ones routinely utilize puzzles in their hiring practices. The rationale for this is the belief that an employee, say a programmer of some sort, is more likely to contribute in a creative, insightful way to the company if they're creative and insightful when presented with a complex puzzle.

Why then are candidates for the presidency never presented with a few simple puzzles to help the electorate gauge their cognitive agility? The same goes for interviewers who ask the same dreary, insipid questions time after time and accept the same dreary, insipid non-answers time after time.

These puzzles shouldn't be difficult since, after all, the primary job of the president is to enforce the Constitution, ensure an honest and open administration, and, in some generalized sense, make things better. For this task, judgment and wisdom are more essential than the ability to solve puzzles. Nevertheless, I think some non-standard questions like the following would help winnow, or at least chasten, some of the candidates.

#### The Puzzles

1. Scaling. Imagine a small state or city with, let's say, a million people and an imaginative and efficient health care program. The program is not necessarily going to work in a vast country with a population that is 300 times as large. Similarly a flourishing small company that expands rapidly often becomes an unwieldy large one. Problems and surprises arise as we move from the small to the large since social phenomena generally do not scale upward in a regular or proportional manner.

A simple, yet abstract problem of this type? How about the following (answers on page 4): A model car, an exact replica of a real one in scale, weight, material, et cetera, is 6 inches (1/2 foot) long, and the real car is 15 feet long, 30 times as long. If the the circumference of a wheel on the model is 3 inches, what is the circumference of a wheel on the real car? If the hood of the model car has an area of 4 square inches, what is the area of the real car's hood? If the model car weighs 4 pounds, what does the real car weigh?

2. Estimating. Proposing any sort of legislation or any action at all requires at least a rough estimate of quantity, costs, benefits, other effects. An ability to gauge them is critical (as is an ability to listen to others' unbiased estimates).

A couple of simple, yet abstract problems of this type? How about the following (hint and answer on page 4): A classic problem: How many piano tuners are there in New York City? And how many times the death toll on Sept. 11 is the annual highway death toll?

3. Sequencing. A president must think about how to gain support for an idea or policy. Some things must be accomplished before other things can be attempted. Legislative backing, popular opinion, domestic and international issues must be dealt with in a reasonable order if an administration is going to be successful. Steps can't be skipped with impunity.

A simple, yet abstract problem of this type? How about the following (answer on page 4): It's very dark and four mountain climbers stand before a very rickety rope bridge that spans a wide chasm. They know the bridge can only safely hold two people and that they possess only one flashlight, which is needed to avoid the holes in the bridge. For various reasons one of the hikers can cross the bridge in 1 minute, another in 2 minutes, a third in 5 minutes and the fourth, who's been injured, in 10 minutes. Alas, when two people walk across the bridge, they can only go as fast as the slower of the two hikers. How can they all cross the bridge in 17 minutes?

4. Calculation. Being able to solve a problem using a bit of algebra, it should go without saying (except to Washington Post columnist Richard Cohen -- link on page 4), can be useful to a politician, whether the issue is taxes, health policy or stock broker commissions.

A simple, yet abstract problem of this type? How about the following, which is not irrelevant to broker commissions (answer below): A 100-pound sack of potatoes is 99 percent water by weight. After staying outdoors for a while, it is found to be only 98 percent water. How much does it weigh now?

5. Deduction, Again, it should go without saying that the ability to make simple deductions is a prerequisite for good decision-making.

A simple, yet abstract problem of this type? How about the following (answer on page 4): Imagine there are three closed boxes, each full of marbles on a table before you. They're labeled "all blue marbles," "all red marbles," and "blue and red marbles." You're told that the labels do describe the contents of the boxes, but all three labels are pasted on the wrong boxes. You may reach into only one box blindfolded and remove only one marble. Which box should you select from to enable you to correctly label the boxes?

Although these problems are much easier than those employers use when hiring entry-level programmers, it would be nice to know that the various candidates, who often are more given to bombast than to logic or evidence, could solve them with ease (although being able to solve them wouldn't be a guarantee of anything). The venue for their solution would be a quiet study with no aides, no pundits, no hot lights, and no intense scrutiny.

What's your guess about how the various candidates would fare with such puzzles? Mine is that a few would find most of the problems trivial, some would have difficulty with them, and the rest wouldn't be sufficiently patient to even try them.

#### The Solutions

*Answers to 1.) 90 inches, 3,600 square inches, 108,000 pounds. (the area increases by a factor of 30^2 (900), and the volume or weight increases by a factor of 30^3 (27,000).*

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.

#### Links

http://www.washingtonpost.com/wp-dyn/content/blog/2006/02/15/BL2006021501989.html

*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.*