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January 4, 2001

Welcome to the Third Millennium.

**Old Challenge.** What would days and seasons be like on a planet shaped like a cube instead of a sphere?

**Answer.** Each face of the cube would be a single time zone, with the sun rising and setting at the same moment everywhere in that face. Also, the height of the sun in the sky, the primary cause of our climates, would be the same everywhere in that face. Therefore, as Rich Mickelsen points out, there would be little variation in temperature, little wind, and "we probably would have suffocated in our own pollution by now."

If the axis of rotation went through the middle of opposite faces, those two whole faces, on the top and bottom, would have one, half-year-long summer day and one, half-year-long winter night, just as at the poles of our round earth. In the four faces on the sides, the sun would pass directly overhead only on the first day of spring and the first day of fall, just as on the equator of our round earth.

If the axis of rotation went though opposite corners, the three faces on top would have winter while the three faces on the bottom had summer.

There would be other major new influences on the climate. As Rich Mickelsen points out, the edges would be **Huge** mountain ranges, thousands of miles high, reaching up into space, dwarfing Mt Everest, which is just about five miles high. Separate civilizations would be isolated around the oceans in the "valleys" at the centers of each face.

**Old Riddle** (Walt Wright). What's next in the sequence 77 49 36 18 ?

**Best Answer** (Rich Mickelsen). 08 = 1x8 (since each number is the product of the digits of the previous one), followed forever by 00.

**New Riddle** (Cihan Altay). What is next in the sequence of hours of the day:

17:14, 12:01, 07:04 ?

**New Challenge** (Nathan Wright). In a sequence of random digits, what is the probability that the first two fives have exactly two digits in between them?

Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for* Flatland *and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2001, Frank Morgan.