Related papers: Smallest Squared Squares

Tile the unit square with $n$ small squares. We determine the minimum of the sum of the side lengths of the $n$ small squares, where the minimum is taken over all tilings of the unit square with $n$ squares.

In this article, we study short intervals that contain another type of "almost square", an integer $n$ which can be factored in two different ways $n = a_1 b_1 = a_2 b_2$ with $a_1, a_2, b_1, b_2$ close to $\sqrt{n}$.

A well known open problem of Meir and Moser asks if the squares of sidelength $1/n$ for $n \geq 2$ can be packed perfectly into a square of area $\sum_{n=2}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}-1$. In this paper we show that for any $1/2…

We consider the problem of packing a large square with nonoverlapping unit squares. Let $W(x)$ be the minimum wasted area when a large square of side length $x$ is packed with unit squares. In Roth and Vaughan's paper that proves the lower…

It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same…

P. J. Federico used the term low-order for perfect squared squares with at most 28 squares in their dissection. In 2010 low-order compound perfect squared squares (CPSSs) were completely enumerated. Up to symmetries of the square and its…

An almost square of type 2 is an integer $n$ that can be factored in two different ways as $n = a_1 b_1 = a_2 b_2$ with $a_1$, $a_2$, $b_1$, $b_2 \approx \sqrt{n}$. In this paper, we shall improve upon previous result on short intervals…

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…

We investigate the number of squares in a very broad family of binary recurrence sequences with $u_{0}=1$. We show that there are at most two distinct squares in such sequences (the best possible result), except under such very special…

We show that there are infinitely many square numbers , which are constrocted by putting two square numbers together , that non of them are divisible by $10$ . We also investigate the interesting properties of some square numbers.

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 cannot be too small.

We study short intervals which contain an ``almost square'', an integer $n$ that can be factored as $n = ab$ with $a$, $b$ close to $\sqrt{n}$. This is related to the problem on distribution of $n^2 \alpha \pmod 1$ and the problem on gaps…

In their 2009 note: \emph{Packing equal squares into a large square}, Chung and Graham proved that the uncovered area of a large square of side length $x$ is $O\left(x^{(3+\sqrt{2})/7}\log x\right)$ after maximum number of non-overlapping…

This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible…

Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits…

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of…

We look at sets of tiles that can tile any region of size greater than 1 on the square grid. This is not the typical tiling question, but relates closely to it and therefore can help solve other tiling problems -- we give an example of…

It is known that $\sum\limits_{i =1}^\infty {1/ i^2}={\pi^2/6}$. Meir and Moser asked what is the smallest $\epsilon$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\ldots$ can be packed into a rectangle of area…

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length…

We show that a large square of sidelength $x$ can be packed by unit squares in a manner so that the wasted space $W(x) = O(x^{3/5})$.