Announcing the Winner of an Intelligencer Contest

First Prize (a free Springer-Verlag book) was awarded to:

Dr. DAVID GAULD University of Auckland

A joint entry from Professor Peter Hilton and Professor Jean Pedersen received honorable mention. Here are the two winning entries together with accompanying letters.

From David Gauld

Dear Editors, I enclose my entry in your 100-word theorem compe- tition as given on page 26 of Volume 5, Number 1. I chose this theorem because I felt it would appear to the seafaring nature of the local residents. Hawaiki, of course, is their mythical land of origin and near there one does indeed find an amphidromic point whose existence is predicted by the theorem.

I must lodge an objection to your blatantly hemis- pherist editorial policy. I believe that the northern and southern hemispheres score about equally in canni- balism in the past century: the only such incidents I am aware of both followed plane crashes, one in each hemisphere.

Best wishes from a (non-cannibalistic) South Sea Is- land.

David Gauld

kei koonei I waenganui ao (

He tikanga

equator to here one crosses a circle near which there is a jump in the number. This is impossible because nearby circles have the same number of cycles. Thus somewhere there is no tidal variation. Go to Hawaiki!

From Peter Hilton and Jean Pedersen

He tikanga

Mehemea, kaahore eetahi whenua i teenei ao haawhe, na reira he waahi, kaahore te tai koorure.

Mehemea he tai koorure kei ngaa waahi katoa, na reira he haerenga a teetahi porowhira kei te whika, he nama a ngaa tai koorure: kei waenganui ao e rua, aa, kei koonei e kaao. Na reira, kei te haerenga mai no te waenganui ao, ka whitia eetahi porowhira kei tata ia he peke teeraa nama. Engari, kaahore e tae na t e mea ka tata ngaa porowhira, ka rite te nama. Na reira, he waahi kaahore te tai koorure. Haere ki Hawaiki!

Rough translation: If there were no land in this hemisphere then at some point there would be no tidal variation. If there were tidal variation everywhere then a single traverse of any circle in the figure results in an integral number of tidal cycles: on the equator there are two cycles and here there are none. Thus on a journey from the

Dear Editors, Here is our recipe for success among the Mathematical Intelligencer's fun-loving South Sea Island cannibals.

First we would say to them:

You know how to construct, using a ruler and compass, regular N-gons for N = 2rNo, where No = 3, 5 or 15. Prince Gauss has proved that others can also be constructed in this way, but only for a very restricted set of values of N.

The we would write on the ceremonial blackboard the attached theorem.

For your interest we refer you to our joint article, "Approximat ing any regular polygon by folding paper; an interplay of geometry, analysis and number theory", Mathematics Magazine, May 1983. But, please don' t give this reference to the hungry cannibals--and don't count the preliminary oral comments, or this ref- erence, in our 100 words!

With our best wishes. Peter Hilton Jean Pedersen

5 0 THE MATHEMATICAL INTELLIGENCER VOL. 5, NO. 4 �9 1983 Springer-Verlag New York

Approximating Regular Polygons

Theorem 1. A s t r a igh t s t r ip of p a p e r m a y be folded to approximate (as accurately as desired) a regular (2 n + 1) -gon. W e p r o v e the typica l case n = 3.

�9 Crea se on l ines (~) , (~) , ~ ) , (~) . . . . to p r o d u c e the ang les ind ica ted .

�9 Cu t across the s t r ip on ( ~ a n d @ .

�9 Fold on (~) , ( ~ , (~) . . . . . ( ~ .

EUREKA! A 9-gon. W h y so? A s s u m e c~ = ~/9 + E. T h e n ~/9 + E + 8x 1

= ~, so Xl = W9 - El8. Similar ly , x2 = ~/9 + E/64 . . . . . a n d xn t e n d s to ~'/9. Af te r d i n n e r w e will s h o w y o u h o w to a p p r o x i m a t e any r egu l a r p o l y g o n b y fo ld ing pape r !

..... ~x, ~ -, x ..... | | @ .... .x, ~x, , .x, .... �9 ,c. . / . . -" 12X, ", z - . . . ; 2X2 / . - - ' " ' ",, ""~

...... . t ,~..) l ~ , , , ~ . . . . . ; / " , . . . . . ~ ( 9 ) ",, ' r . . . . . o. (a rb i t ra ry ) ~ N ""-. .~-.f. . . . . Xi v "N

Approx imat ing Regular Polygons

THE MATHEMATICAL 1NTELLIGENCER VOL. 5, NO. 4, 1983 51

Who Is It?

52 THE I~.ATHEMATICAL INTELLIGENCER VOL. 5, NO. 4 �9 1983 Springer-Verlag New York