The three squares in a rectangle problem asks for the maximum sum of the side lengths of three squares contained in a rectangle, where and the interiors of the squares are pairwise disjoint. The squares may have different sizes and may undergo arbitrary rotation. Yang (2026) determined the maximum to be

In particular, the maximum is 3/2 in a unit square and 2 in a rectangle. The latter value is attained by one unit square and two squares of side length 1/2. For , three unit squares attain the maximum 3. The upper bounds allow all orientations, even though these examples have sides parallel to the boundary.

See also

Rectangle, Square Packing, Unit Square

Explore with Wolfram|Alpha

References

Yang, H. "Three Squares in a Rectangle." 27 Aug 2026. https://arxiv.org/abs/2608.13595.

Cite this as:

Weisstein, Eric W. "Three Squares in a Rectangle Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ThreeSquaresinaRectangleProblem.html

Subject classifications