Math Chat

Prime Number Digits

Old kings and queens challenge question

A magician takes a stack of eight cards (four kings and four queens, face down, in some order), turns the top one face up, moves the next one to the bottom of the stack, turns the next one face up, moves the next one to the bottom, and so on.

To deal out an alternating sequence of kings and queens, how must he arrange the cards beforehand? Suppose instead he has some number of cards, each with a letter of the alphabet. Can you arrange them to spell out some word or phrase, which when dealt out spell out another word or phrase?

Answer

From the top, start with 2 kings, 1 queen, 2 kings, 3 queens. (Try it, then impress some friends.) You can find this answer by running the whole process backwards.

With letters instead of royalty, Brennie Morgan found some 5-letter examples: beats becomes baste, heats haste, and peats paste (not to mention error still results in error).

Chuck Gahr found ad rem yeilds armed. A prize awaits anyone who can find an example with 6 or more letters.

New prime number

A new largest prime number has just been discovered by David Slowinski and Paul Gage at Silicon Graphics using a Cray T94 supercomputer. Like most known huge primes, it is a "Mersenne" number, obtained by multiplying 2 by itself a huge number of times and then subtracting 1. The new prime is 21,257,787 minus 1, a number 378,632 digits long, more than 100 single-spaced typewritten pages. The previous record holder was 2859,433 minus 1, discovered in 1994.

New challenge question

For this new prime number, what is the last digit? the second to the last digit? the first digit? the second digit?

*Send answers, comments, and new questions to:

Math Chat

Bronfman Science Center

Williams College

Williamstown, MA 01267

or by e-mail to:

Frank.Morgan@williams.edu.

The best submissions will receive a copy of the book Flatland, which helps explain higher dimensions. The challenge-question answer and winner will be announced in a future column.

Math chat is a biweekly column, based on a live call-in show on local-access television in Williamstown, Massachusetts.