48 Normat 55:1, 48 (2007)
Summary in English
Ivar Farup, The Piling of Books (Nor-
wegian).
Give n an inexhaustible supply of books.
Is it possible to pile them up on a table
such that they extend arbritarily far
from the e dge of the table? In the art-
icle this is shown to be the case through
a simple construction which ought to be
well-known.
Ragnar Solvang, Leibniz triangle
(Norwegian).
Pascals triangle is we ll-known. In this
article another triangular exhibition of
numbers, this one with fractional entries
and introduced by Leibniz will be stud-
ied. In particular the sums of rows and
columns will be computed.
Ulf Persson, The Mercator projecton
(Swedish).
The Mercator projection is usually
presented severely truncated in atlases,
for obvious reas ons as the earth is
mapped onto an infinite strip. Surpris-
ingly though a fairly modest truncation
accomodates all of the earth except two
coin-sized regions at the poles. In fact
such a truncated map will be presented
with the upper and lower extremes of
the rec tangle are scaled 1:1.
Jorge Nuno Silva, Mathematics and
Games:Hex .
The well-known game of hex is be-
ing studied. This is a game in which
draw s are not possible. It is shown that
this property is ac tually equivalent to
Brouwers celebrated fixpoint theorem.
Anders Thorup, The Josefus per-
mutation (Danish)
It is well-known that any permutation
can be presented as a composition of
cycles. In fact every permutation has a
unique representation, up to order, as a
product of disjoint (and hence commut-
ing) cycles. It turns out that the com-
position c
n
c
n−1
. . . c
2
where c
k
=
(1 2 3 . . . k) has a nice and eas-
ily found such representation, but sur-
prisingly if the order of the cycles is re-
versed the problem becomes almost in-
tractable. The elementary nature of the
problem gives a good exposure to stu-
dents, and it has in fact been studied by
students over the years. The article is in
the nature of a progress report, and the
cases of n = 2
m
, 2
m
− 1 are treated in
detail. Surprising connections to other
parts of combinatorics are highlighted.
Kent Holing, Pythagoreiske tripler
(Norwegian)
Some generalizations of previous prob-
lems are discussed in the context of py-
thagorean triplets.
