Minimally Squared Rectangles

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Divide a rectangle of size into the minimal number of squares. For a rectangle of size 17×19, the solution (with nine squares) is nontrivial to find. This Demonstration gives precalculated solutions for rectangles up to size 380×380.

Contributed by: Ed Pegg Jr (March 2013)
Based on data compiled by Bertram Felgenhauer and Ed Pegg Jr
Open content licensed under CC BY-NC-SA



Let represent the minimal number of squares for an rectangle. Conjecture: for all and .

These solutions were compiled in the mistaken belief that a counterexample would be found. So far, the conjecture holds true for . Solutions are found with a method involving Young tableaux; a single solution can require hours to find [1].

Some rectangles can be divided into squares of different sizes. These are known as perfect rectangles, and are much easier to find [2].


[1] B. Felgenhauer. "Filling Rectangles with Integer-Sided Squares." (Mar 17, 2013)

[2] S. Anderson. "Squaring.Net 2013." (Mar 26, 2013)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.