Math and sicence puzzles have amused and befuddled man for centuries. Many of us delight in the challenge of pitting our wits aginst historic brain busters.

The 20 puzzles here have stumped the best thinkers in the world-it took an Einstein or a Newton to figure our some of them. Go ahead. Test yourself. The answers immeditately follow the questions.1. What's the largest number? The smallest?

2. Suppose you just tossed a "fair" coin three times and got all heads. You're more likely to get a tails on the next try, right?

3. If you just planted a row of five fence posts, with 10 feet separating each successive pair, how long is the finished fence?

4. Do you weigh more on an elevator that's going up or down?

5. Is 1,000 ever equal to eight?

6. Are any of the following not equal to zero? Two times zero, zero divided by two, two divided by zero?

7. Would you rather have $1 million or a penny doubled every day for a month (in other words, on the first day you get a penny, on the second day you get 2 cents, on the third day you receive a total of 4 cents, etc.)? Does your answer depend on which month it is?

8. Can you tell - without using pencil and paper - if the following number is divisible evenly by 9? 210,121,021,212,012.

9. Imagine you could ride along on a speeding light beam. What time would your watch show three "Earth days" from now?

10. When a tree falls in a forest but nobody hears it, does it make a sound?

11. What's wrong with the following logic? That bird is a bald eagle. But bald eagles are disappearing. Therefore, that bird must be disappearing.

12. Let's say you're running fast toward a destination. Now there's a point halfway there that you still have to reach. But once you reach that point, there's another halfway point closer in that you still have to get to, and so forth. So you can never quite reach your destination, right?

13. Can you compute the sum of the following infinite (in other words, unending series? 1/2 + 1/4 + 1/8 + 1/16 + 1/32. . . .

14. Try this logic puzzle: In a small village there's a single barber. He claims, "I shave all the village men who don't shave themselves. But of course, I don't shave any who do shave themselves." Given what the barber says, is it possible to determine who shaves him?

15. Would an apple or a bowling ball fall faster if dropped?

16. Is there a number that when divided by itself gives the same answer as when multiplied by itself?

17. What does Einstein's famous equation E = mc2 (involving energy, mass and the speed of light c) say, fundamentally, about matter and energy?

18. Suppose you're at a party with 28 strangers when someone offers to bet you $10 that at least two people in the room share the same birthday. Assuming nobody is cheating (in other words, neither of you has inside information about the partygoers' birthdays), should you accept the bet?

19. Let's say you're making a map of the United States or some other part of the world. How many different colors would you need to be sure that no two adjacent states or countries (in other words, ones sharing a common boundary - single points don't count) have the same color?

20. If you journeyed into space in a high-speed rocket ship while your twin stayed back on Earth, which of you would be older upon your return?

ANSWERS

1. There is no largest or smallest number, because you can always add or subtract one from any given number to get an even bigger or smaller number.

2. For a "fair" coin, the odds are still 50-50 on any given toss, no matter what happened in the past. Luck has no memory.

3. 40 feet. If you said 50 feet, you erroneously counted the first post - a common "boundary" problem that crops up in math and science.

4. Slightly more when the elevator starts up, less when it starts down. But once constant speed is attained, your weight is "normal."

5. Yes. In computer binary language, which employs base 2, 1 is one, 10 is two, 100 is four and 1,000 is eight.

6. Two divided by zero. Dividing by zero would yield an answer so big our number system can't handle it - infinity. So such divisions are not permitted in ordinary arithmetic.

7. You should choose the doubling, regardless of the month. A 28-day February would yield $1.3 million; a 31-day month would bring $11 million.

8. Because the sum of the digits - 18 - is evenly divisible by nine, so is the original number. But don't try this using sevens or any other digit.

9. The same time as when you left. Time comes to a standstill when you travel at the speed of light, declared Albert Einstein.

10. That depends on your definition of "sound." If you mean waves in the air, the answer is yes. If you mean the subjective experience of hearing, the forest is silent. Defining one's terms precisely in an argument is at the heart of much of philosophy.

11. You shift the meaning of "disappearing" in midstream here - a logical no-no.

12. This puzzle, first posed thousands of years ago by the Greeks, took many centuries to clear up: Though you do indeed travel through an unending sequence of midpoints, the closer you get to your goal, the faster you pass these midpoints until the time interval between them becomes, finally, infinitesimally small - you're there!

13. It's one. If you can see how this parallels the problem of number 12, you may have a future in mathematics.

14. No. His two statements can't both be true at the same time. This type of contradiction shook the foundations of mathematics early in this century.

15. If you discount air resistance (a big "if"), all objects fall at the same rate.

16. There are actually two such numbers: positive 1 and negative 1.

17. That the two are really one and the same, and that it doesn't take much matter to yield enormous amounts of energy (as witness an atomic bomb).

18. Only if you're really a gambler at heart. Odds are about 2-to-1 you'll lose your 10 bucks.

19. Four different colors will do the job, whether you're drawing up the United States or some other area of the world. While this has been known in practice by map makers for a long time, not until 1976 was it finally proved (using some powerful computers).

20. Your twin would be older, because very great speeds and accelerations slow clocks. Actually, centuries may have passed on Earth before you return.