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January 18, 2001

The joint mathematics meetings in New Orleans last week included a talk by MacArthur "Genius" award winner Jeffrey Weeks on the geometry and topology of the universe. If for example the universe were a big torus or "doughnut," we might see the same radiation from afar in opposite directions. The upcoming Microwave Anisotropy Probe (MAP) may discover such patterns.

**Old 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?

**Answer** 8.1%, computed as the probability that the next digit is not a five (.9) times the probability that the next digit is not a five (.9) times the probability that the next digit is a five (.1), for a product of .9 x .9 x .1 = .081 = 8.1%. Al Zimmermann writes:

"Having insomnia tonight, I wrote a program to verify this. I generated 1,000,000 sequences of random digits until each one had produced two fives. Here are the numbers of digits between the two fives:

Number of non-5's | Number of times | Probability |

0 | 99911 | 0.0999 |

1 | 89819 | 0.0898 |

2 | 80853 | 0.0809 |

3 | 72756 | 0.0728 |

4 | 65613 | 0.0656 |

5 | 59141 | 0.0591 |

6 | 53179 | 0.0532 |

7 | 47874 | 0.0479 |

8 | 43413 | 0.0434 |

9 | 39197 | 0.0392 |

10 | 34654 | 0.0347 |

The program also calculated the average number of non-fives between the two fives. This came out to 9.003, which compares nicely to the theoretical value of

which turns out to be exactly 9."

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

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

**Answer.** 02:06. On a digital clock 17 consists of 5 lighted segments (2 for the 1 and 3 for the 7), 14 consists of 6, 12 consists of 7, 01 consists of 8, 07 consists of 9, 04 consists of 10, 02 consists of 11, and 06 consists of 12. (When there are more than one possibility, use the numerically smallest.)

**New Challenge** (Walter Wright). Consider the sequence: 3 1 2 1 3 2. It consists of a pair of 1's separated by one other number, a pair of 2's separated by two other numbers, and a pair of 3's separated by three other numbers. Can you find a similar sequence with a pair of 4's too? More?

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.