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April/May 2000 Issue

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Today's Editors: Jane Heckler and Dan Kalman

In this Issue.....

Final Issue for the Semester
Four 9's Puzzle
More on that Induction Proof
Interesting Math Web Pages
Calendar
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Final Issue for the Semester

This will be the final Liaisons Newsletter for the 99-2000 academic year. We would like to take this opportunity to thank all liaisons for their participation in the program, and we hope you will want to continue to participate next year. Enjoy the summer!

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The Four 9's Puzzle

Have you ever heard of the four 9's puzzle? The object is to represent every integer in some range (say 1 to 99) using four 9's and operations symbols. For example, 2 can be written as 9/9 + 9/9, 3 is (9+9+9)/9, and so on. A graduate student at American University, Terry Moore, recently observed a complete solution of the four 9's problem by cleverly modifying a solution to a similar problem. To illustrate Moore's solution, here is the representation of the number 4:

log _ _ log _ _ _ _ 9 = 4

(.9 + .9) -/ -/ -/ -/ 9

_

Note that .9 is meant to represent the infinite repeating decimal .99999... and the characters following the second log are supposed to be radical symbols, so that the base of the second log is four nested square roots of 9, or the 16th root of 9. Putting all that together, the expression on the left is the log base 2 of the log base 16th-root-of-9 of 9. The generalization of this to any n simply requires repeating more or fewer of the radical symbols. In general, the same expression with n radical symbols is equal to the number n.

Moore obtained this formula by modifying a marvelous creation of Verner Hoggatt, notable as founder of the Fibonacci Quarterly, among other things. Hoggatt's formula expresses 4 as

log log _ _ _ _ 9

(0+1+2+3+4)/5 -/ -/ -/ -/ (-6+7+8)

In this way, Hoggatt was able to express any integer using all of the ten decimal digits, once each, in order! An account of this discovery, and much more about Hoggatt, can be found in the article by Howard Eves, "Hail to thee, blithe spirit!", The Fibonacci Quarterly, 19 (1981) 193-196.

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More on that Induction Proof
By Dan Kalman

In the last newsletter, I reported on the proof that Arthur Benjamin presented at the January meetings. Then, in a follow up message, an error in my presentation of the proof was corrected, and an extension was given to a more general version. In retrospect, I now believe that Benjamin's original presentation was of the more general version. In any case, Roger Nelsen, the Liaison at Lewis and Clark College, has written to point out that the general version of the induction proof first appeared in an article by Solomon Golomb entitled "Checker Boards and Polyominoes", American Mathematical Monthly, Vol 61, No. 10 (December 1954) pp. 675-682. The full text of that article can be found online at JSTOR. See

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Interesting Math Web Pages

Francis Su at Harvey Mudd College has a neat site devoted to what he calls math fun facts. Visit the site for tidbits to enliven your math classes: http://www.math.hmc.edu/funfacts/

Another interesting site is "The Number Years"

http://www.jquinn.oxy.edu/numberyears/

This site provides a series of game-show like activities for recreational math events, such as math club meetings or mathfests.

The following web site contains an annotated list of links to web pages concerning the history of mathematics:

http://www.sub.uni-goettingen.de/ssgfi/math/subject/math_msc01_on_en.html

Each listing includes some information about the site to which it links. Some examples include:

Mathematical Quotations Server:

http://math.furman.edu/~mwoodard/mqs/mquot.shtml

Mathematicians of the African Diaspora

Women Mathematicians

http://www.agnesscott.edu/lriddle/women/women.htm

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Calendar

June 8...Early Bird registration deadline for Mathfest 2000
(www.maa.org/students/students_index.html)
(www.maa.org/awards/morgan_nom.html)
July 12-15...Intl Derive/TI-92 Conference
(www.cms.livjm.ac.uk/derive2k)
August 1-2...MAA Short Course, "Introduction to Error Correcting Codes"
(www.maa.org/meetings/short_course00.html)
August 3...Teaching Workshop for Grad Students & New Faculty, Mathfest
(www.maa.org/meetings/teaching_workshop00.html)
August 3-5...Mathfest 2000
(www.maa.org/meetings/mathfest00_frontpage.html)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The Liaisons Newsletter is produced at the headquarters office of the MAA. Comments, suggestions, and questions are welcome. Please direct them to any of the following:

Jane Heckler
Senior Assistant for Programs
jheckler@maa.org

Dan Kalman
Member, Subcommittee on MAA/Departmental Liaisons
kalman@email.cas.american.edu

John Petro
Chair, Committee on MAA/Departmental Liaisons
john.petro@wmich.edu

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