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Amicable Heron triangles
Iwan Praton, Nart Shalqini
A Heron triangle is a triangle whose side lengths and area are integers. Two Heron triangles are amicable if the perimeter of one is the area of the other. We show, using elementar…
Maximum tilings with the minimal tile property
Iwan Praton
A tiling of the unit square is an MTP tiling if the smallest tile can tile all the other tiles. We look at the function , where is the side length of the…
Tilings with the Minimal Tile Property
Iwan Praton
A square tiling of the unit square is said to have the minimal tile property if the smallest tile can tile all the other tiles. We show that in such a tiling, the smallest tile can…
Minimal tilings of a unit square
Iwan Praton
Tile the unit square with small squares. We determine the minimum of the sum of the side lengths of the small squares, where the minimum is taken over all tilings of the un…
Tiling a unit square with 8 squares
Iwan Praton
Put n nonoverlapping squares inside the unit square. Let f(n) and g(n) denote the maximum values of the sum of the edge lengths of the n small squares, where in the case of f(n) th…
The Erdos and Campbell-Staton conjectures about square packing
Iwan Praton
Put n open non-overlapping squares inside a unit square, and let f(n) denote the maximum possible value of the sum of the side lengths of the n squares. Campbell and Staton, buildi…