1. The smallest number of soldering points remains eight no matter how wires are bent. Because an odd number of edges meet at each corner of a cube, every point requires soldering.

2. Do not put Descartes before the horse.

3. You can evade check forever. Head toward the board's center, always moving your king to a color opposite to that of the knight. Since a knight changes the color of its cell at every move, whenever the king is on a color different from the knight's, no knight move can check the king. Your only danger lies in being trapped in a corner where you can be forced to move diagonally and be checked by the knight's next move.

4. The highest constant is 36 [see Figure 41].

5. One answer: "The value of n is less than one million."

6. He makes twice as much money.

Figure 41

Figure 41

Answer to the card problem

7. The black curve divides the plane into a number of regions. Trace a ro>u n-dl trip along the red curve and it is obvious that every region you enter you must also leave or you will never get back to tvhere you started. Since each entrance and exit is a pair of c ros.sing points, the total number of such spots must be even.

9. Two and a half times as high.

10. With one unit side as base and the other unit side free to rotate [see Figure 42], the triangle's area is greatest when the altitude is maximum. The third side will then be the square rooL of 2.

12. Only two of the eight possible combinations of crossings create a knot, making the probability of a knot 2/8 = 1/4.

13. House.

14. The first number, 1,324, raised to any power must end in 6 or 4. The other two numbers, 731 and 1,961, raised to any power must end in 1. Since no number ending in 6 or 4, added to a number ending in 1, can produce a number ending in 1, the equation has no solution.

15. Underground.

16. One proof that the probability is 1/2: Suppose the man has a Doppelgänger directly opposite him on the other side of the Pentagon's center and the same distance away. If either man sees three sides, his double must see only two. Since there is an equal probability that either man is at either spot, the probability is 1/2 that he will see three sides.

18. The four possible true-false combinations for the two statements are TT, TF, FT and FF. The first is eliminated be

Figure 42

Arswer to the triangle problem cause we were told that one statement is false. The second and third are eliminated because in each case, if one person lied, the other cannot have spoken truly. Therefore both lied. The boy has red hair, the girl black hair.

19. Figure 43 shows the two solutions.

20. If ABCD= 1,234, it is impossible to obtain a sum as large as 12,300. If ABCD = 3,456, it is impossible to obtain a sum as small as 12,300. Therefore ABCD = 2,345, from which it is easy to determine that the four dots stand for 4,523.

21. It is well known that every prime greater than 3 is one more or one less than a multiple of 6. It is easy to see that every number of the form 6n ± 1 must fall on the same diagonal, therefore the diagonal is certain to catch every prime.

22. Any two positive integers.

23. There are three permissible combinations of true and false for the three statements: TFF, FTF and FFT. The only noncontradictory combination is FTF, which means that Feemster owns no books at all.

24. Each month has 11 ambiguous dates (a date such as 8/8/71 is not ambiguous), making 132 in all.

25. A square manhole cover, turned on edge, could slip through its hole and fall into the sewer.

26. Ten digits can be permuted in 101 = 3,628,800 different ways. A 10-digit number cannot start with zero, so that we must subtract 3,628,800/10 = 362,880 to obtain the answer: 3,265,920.

27. No, because after 72 hours it would have been midnight again.

28. "To be or not to be, that is the question."

29. Instead of inscribing the hexagon as shown, turn it to the position shown in Figure 44. The grey lines divide the larger hexagon into 24 congruent triangles, 18 of which form the smaller hexagon. The ratio of areas is 18 : 24 = 3 : 4, and so if the smaller hexagon has an area of three, the larger one has an area of four.

30. Smith was born in 1892. He was 44 in 442= 1936.

31. I write "10." This is any base written in that base system's notation.

33. Twenty white cubes can abut the grey cube. Arrange seven white cubes as shown in Figure 45. The grey cube goes on top as. indicated. Seven more white cubes, in the same pattern and position as the first layer, form layer No. 3. In between layer INo. 1 and layer No. 3 six more white cubes can be placed: two on each of two opposite sides of the grey cube and single cubes on the remaining two sides.

Figure 45


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Arrangement of the cubes

Arrangement of the cubes

34. The consecutive letters RSTU will spell "rust" or "ruts."

35. He ties one end of the rope to the tree at the edge of the lake, walks around the lake holding the other end of the rope and ties that end to the same tree. The doubled rope is now firmly stretched between the two trees, making it easy for the man to pull himself through the water, by means of the rope, to the island.

36. The dog can be at any point between the boy and the girl, facing either way. Proof: At the end of one hour, place the dog anywhere between the boy and the girl, facing in either direction. Time-reverse all motions and the three will return at the same instant to the starting point.

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