The Heilbronn Problem for Squares

The following pictures show n points inside a unit square so that the area A of the smallest triangle formed by these points is maximized. The smallest area triangles are shown.

3.

A = 1/2 = .500

Trivial.

One of an infinite family of solutions.

Symmetric about a diagonal.

4.

A = 1/2 = .500

Trivial.

Completely symmetric.

5.

A = √3 / 9 = .192+

Proved optimal by Yang Lu, Zhang Jingzhong, and Zeng Zhenbing, 1991.

Symmetric about a diagonal.

6.

A = 1/8 = .125

Proved optimal by A. Dress, L. Yang, and Z. B. Zeng, 1995.

One of an infinite family of solutions.

Vertically and horizontally symmetric.

7.

A = .0838+

Found by F. Comellas and J. Yebra, December 2001.

Proved optimal by Zhenbing Chen and Liangyu Chen, 2011.

Not symmetric.

8.

A = (√13–1) / 36 = .0723+

Found by F. Comellas and J. Yebra, December 2001.

Proved optimal by L. Dehbi and Z. Zeng in 2022.

180^o Rotationally symmetric.

9.

A = (9√65–55) / 320 = .0548+

Found by F. Comellas and J. Yebra, December 2001.

Symmetric about a diagonal.

10.

A = .0465+

Found by F. Comellas and J. Yebra, December 2001.

Symmetric about both diagonals.

11.

A = 1/27 = .0370+

Found by Michael Goldberg, 1972.

Horizontally symmetric.

12.

A = .0325+

Found by F. Comellas and J. Yebra, December 2001.

Completely symmetric.

13.

A = .0270+

Found by Peter Karpov, August 2011.

Not symmetric.

14.

A = .0243+

Found by Mark Beyleveld, August 2006.

Symmetries of a rectangle.

15.

A = .0211+

Found by Peter Karpov, August 2011.

Not symmetric.

16.

A = 7 / 341 = .0205+

Found by Mark Beyleveld, August 2006.

180^o Rotationally symmetric.

More information is available at Mathworld, and the article by Comellas and Yebra.