Summaries 1/2007

Ivar Farup
The Piling of Books (Norwegian).
Given an inexhaustible supply of books. Is it possible to pile them up on a table such that they extend arbritarily far from the edge of the table? In the article 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 well-known. In this article another triangular exhibition of numbers, this one with fractional entries and introduced by Leibniz will be studied. 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 reasons as the earth is mapped onto an infinite strip. Surprisingly 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 rectangle are scaled 1:1.

Jorge Nuno Silva
Mathematics and Games:Hex.
The well-known game of hex is being studied. This is a game in which draws are not possible. It is shown that this property is actually equivalent to Brouwers celebrated fixpoint theorem.

Anders Thorup
The Josefus permutation (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 commuting) cycles. It turns out that the composition c_n c_{n-1}… c_2 where c_k=(1 2 3 … k) has a nice and easily found such representation, but surprisingly if the order of the cycles is reversed the problem becomes almost intractable. The elementary nature of the problem gives a good exposure to students, 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 problems are discussed in the context of pythagorean triplets.