2
votes
1answer
36 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
2
votes
1answer
166 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
14
votes
3answers
1k views

Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.

Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that ...
1
vote
1answer
93 views

Tiling a minimal perimeter region with $n$ unit squares

Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square. Is ...