Problem 106

Erdős Problem #106 [status: falsifiable; formalized: no]. Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the squares. Is $f(k^2+1)=k$? Current best: Hal\'{a}sz [Ha84] gives a construction that shows $f(k^2+2)\geq k+\frac{1}{k+1}$, and in general, for any $c\geq 1$,$$f(k^2+2c+1)\geq k+\frac{c}{k}$$and$$f(k^2+2c)\geq k+\frac{c}{k+1}.$$Hal\'{a}sz also considers the variants where we replace a square by a parallelogram or triangle. Erd\H{o}s and Soifer [ErSo95] and Campbell and Staton [CaSt05] have conjectured that, in general, for any integer $-k<c<k$, $f(k^2+2c+1)=k+\frac{c}{k}$, and proved the corresponding lower bound. Prize: no. Tags: geometry.