104 Normat 52:1, 104 (2004)

Summary in English

Hans Chr. Hansen, Number magic

with squares (Danish.) This article was

inspired by a problem from the web

page of the Danish Assoc iation of Math-

ematics Teachers. It was proposed as an

exercise for school pupils learning sub-

traction. Start with a square and one

integer in each corner. Form a smaller

square with an integer in each corner by

placing the diﬀerence of each adjacent

pair of integers in the middle of the cor-

responding side of the original square,

and connecting the new quadruple of

integers by line segments. Re peat this

step. The problem is to discover how

many squares have to be drawn before

the integers become zero at all four cor-

ners.

The author treats the more ge neral

problem of a regular polygon by a mix-

ture of theory and numerical experi-

ment. The case of a square is studied in

detail. Among the results for this case

are that the number of steps to reach a

square with zero at all corners is always

ﬁnite, but can be arbitrarily large.

D. G. Rogers, Pythagoras framed, cut

up Liu Hui (translated into Norwegian

and adapted by Christoph Kirfel.) This

is an exposition of some dissection ar-

guments in plane geometry, inspired by

an ancient Chinese geometrical tradi-

tion. The eminent Chinese mathemati-

cian Liu Hui wrote a commentary in

263 on the older mathematical treatise

Jiu Zhang Suan Shu (Nine Chapters on

the Mathematical Art). The author dis-

cusses some dissection arguments from

this commentary, dealing with formulas

for the diameter of the c ircle inscribed

in a right triangle, and extends the ar-

guments.

Audun Holme, Arabic mathematics 2

(Norwegian.) This is the second of two

articles on Arabic mathematics, the ﬁrst

one appeared in the previous issue. The

author discusses the lives and work of

some important mathematicians from

the Islamic world in the period from

the tenth century to the Spanish Recon-

quista. They made remarkable contribu-

tions to algebra, number theory, geom-

etry, astronomy and the art of numer-

ical calculation. And Nasir al-Din al-

Tusi (1201–1274) founded trigonometry.

Nasir lived in the turbulent time of the

Mongol invasions, and became an ad-

viser to Hulagu Khan.

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