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Puzzling

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Once again, for your edification and amusement, 10 puzzles. You'll find the answers on page 49 of this issue of the Journal. Let's start with the lovely "checkershadow" by Edward Adelson (who kindly refuses to copyright his highly original illusions):

1. In Figure 1, which square is darker, A or B?

2. Easier than it looks: What's the product of two primes whose sum is 7,921?

3. In Figure 2, three angles, X, Y and Z, are drawn on a square grid. What's their sum?

4. When philosopher Immanuel Kant forgot to wind his clock, he went to visit his old pal Schmidt, who had a clock on his wall. After a long chat — perhaps on the nature of time — he walked home by the same route he'd taken and immediately set his own clock correctly. How did he know what time it was?

5. Where was this photo (Figure 3) taken?

6. What's the sum of the infinite series 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ... ?

7. A termite starts on the corner of a 3-by-3-inch cube, as shown in Figure 4 , boring her way through the middle of each cube in turn, never returning to a cube she's already visited. Can or can't she end in the center cube after boring through each of the other 26 cubes? (Hint: think "parity.")

8. Lewis Carroll invented "doublets" in which one word is changed to another one letter at a time (e.g. APE to APT to OPT to OAT to MAT to MAN). Can you get from GRASS to GREEN in seven steps? How about WINTER to SUMMER in eight?

9. Flip four coins. What are the odds that the number of heads will exceed the number of tails?

10. Who claimed to rob banks "because that's where the money is"?

Answers

1. They're the same. Cut out or fold over to see for yourself. (Weird, huh?)

2. For the sum of two numbers to be odd, one has to be even and one odd. Since 2 is the only even prime, the other number is 7919, so their product is 15,838. (Incidentally, 2 and 7,919 are the first and 100th primes.)

3. Add another row of squares at the top and draw isosceles triangle ABC. ACB is a right angle (by symmetry), so CAB is 45 degrees, i.e. X. Mirror angle Z to see that X + Y + Z = 180 degrees. (See Figure 5.)

4. Kant wound his own clock, noting the (incorrect) time when he left. When he returned, h e subtracted from the elapsed time the start and end times of his visit at Schmidt's, giving him his total walking time. Adding half of that to the time he left Schmidt's gives him his return time.

5. Samoa County Park.

6. This is known as Grandi's series after Italian mathematician-philosopher Guido Grandi (1671-1742). The answer is: It depends how you group the numbers. Grouping them (1 - 1) + (1 - 1) + (1 -1)...gives you

zero. But you can just as well group them as 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) ... which sums to 1.

7. No, she can't. Imagine the cubes as a 3D checkerboard, black and white cubes alternating. If there are 14 black cubes, there will be 13 white ones, and the termite's route will alternate through black and white cubes, starting and ending in a black cube. But if the starting corner cube is black, the center cube has to be one of the white ones. So the challenge is impossible. (Similarly you can show that a rook can't start at one corner of a chessboard and end at the opposite corner, passing through every square en route just once.)

8. GRASS to CRASS to CRESS to TRESS to TREES to TREED to GREED to GREEN. (If you allow the archaic "grees," you can four-step it: GRASS-GRAYS-GREYS-GREES-GREEN.) WINTER to WINDER to WANDER to WARDER to HARDER to HARMER to HAMMER to HUMMER to SUMMER.

9. 5/16. Of 16 possibilities (2 x 2 x 2 x 2), five will give more heads than tails (HHHH, HHHT, HHTH, HTHH and THHH).

10. Not bank robber Willie Sutton. An unknown journalist who interviewed Sutton in jail came up with the line for his story. Later, Sutton told a reporter, "Why did I rob banks? Because I enjoyed it. I loved it. I was more alive when I was inside a bank, robbing it, than at any other time in my life."

Barry Evans' (barryevans9@yahoo.com) addiction to puzzles is due to Martin Gardner, whose "Mathematical Games" column in Scientific American ran from 1956 to 1981.


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