Problem

ABCD DCB A

ABCD are four consecutive digits in increasing order. DCBA are the same ffo ut in decreasing order. The four dots represent the same four digits in an unknown order. If the sum is 12,300» what rrumber is represented by the four dots? (W. T. Williams and G-. H. Savage, The Strand Problems Book.)

Problem 21: A "primeval snake" is formed by writing the positive integers consecutively along a snaky path [see Figure 38]. If contimaed upward to infinity, every prime number will fall on the same diagonal line. Explain. (M. G.)

Figure 38

51 52 55

39 40 43

26 29 30

i yj

I KJ

15 16 19

The primeval snake

Problem 22: Find two positive integers, x and y, such that the product of their greatest common divisor and their lowest common multiple is xy.

Problem 23: "Feemster owns more than a thousand books," said Albert.

"He does not," said George. "He owns fewer than that."

"Surely he owns at least one book," said Henrietta. If only one statement is true, how many books does Feemster own?

Problem 24: In this country a date such as July 4, 1971, is often written 7/4/71, but in other countries the month is given second and the same date is written 4/7/71. If you do not know which system is being used, how many dates in a year are ambiguous in this two-slash notation? (David L. Silverman.)

Problem 25: Why are manhole covers circular instead of square?

Problem 26: How many different 10-digit numbers, such as 7,829,034,651, can be written by using all 10 digits? Numbers starting with zero are excluded.

Problem 27: Many years ago, on a sultry July night in Omaha, it was raining heavily at midnight. Is it possible that 72 hours later the weather in Omaha was sunny?

Problem 28: What well-known quotation is expressed by this statement in symbolic logic?

Problem 29: Regular hexagons are inscribed in and circumscribed outside a circle [see Figure 39]. If the smaller hexagon has an area of three square units, what is the area of the larger hexagon? (Charles W. Trigg, Mathematical Quickies.)

Figure 39

Figure 39

Hexagon problem

ProW-em 30: "I was n years old in the year n2," said Smith in 1971. When was he born?

Problem 31: If you think of any base greater than 2 for a number system, I can immediately write down the base without asking you a question. How can I do this? (Fred Schuh, The Master Book of Mathematical Recreation.)

Problem 32: What was the name of the Secretary General of the United Nations 35 years ago?

Problem 33: You have one red cube and a supply of white cubes all the same size as the red one. What is the largest number of white cubes that can be placed so that they all abut the red cube, that is, a positive-area portion of a face of each white cube is pressed flat against a positive-area portion of a face of the red cube. Touching at corner points or along edges does not count. (M. G.)

Problem 34: What four consecutive letters of the alphabet can be arranged to spell a familiar four-letter word? (Murray R. Pearce, Word Ways, February 1971.)

Problem 35: Figure 40 is a diagram of a deep circular lake, 300 yards in diameter, with a small island at the center. The two black spots are trees. A man who cannot swim has a rope a few yards longer than 300 yards. How does he use it as a means of getting to the island?

Figure 40

Figure 40

Lake, island, and trees

Problem 36: A boy, a girl and a dog are at the same spot on a straight road. The boy and the girl walk forward—the boy at four miles per hour, the girl at three miles per hour. As they proceed the dog trots back and forth between them at 10 miles per hour. Assume that each reversal of its direction is instantaneous. An hour later, where is the dog and which way is it facing? (A. K. Austin.)

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