On some new simple perfect squared squares

Simple Perfect Squared Squares and 2 × 1 Squared Rectangles of Orders 21 to 24

Simple perfect squared squares and 2 * 1 squared rectangles of order 26

In this note tables of all simple perfect squared squares and simple 2 × 1 perfect squared rectangles of order 26 are presented.

Simple perfect squared square of lowest order

This note is to report the existence of a simple perfect dissection of a square into 21 unequal squares.

Perfect Matchings and Perfect Squares

In 1961, P.W. Kasteleyn enumerated the domino tilings of a 2n × 2n chessboard. His answer was always a square or double a square (we call such a number "squarish"), but he did not provide a combinatorial explanation for this. In the present thesis, we prove by mostly combinatorial arguments that the number of matchings of a large class of graphs with 4-fold rotational symmetry is squarish; our ...

Some new perfect Steiner triple systems

In a Steiner triple system STS(v)=(V,B), for each pair {a, b} ⊂ V , the cycle graph Ga,b can be defined as follows. The vertices of Ga,b are V \{a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25 and 33. We ...