Guillotine cutting is asymptotically optimal for packing consecutive squares

Abstract More than half a century ago Martin Gardner popularized a question leading to the benchmark problem of determining the minimum side length of a square into which the squares of sizes %$1,2,\dots ,n%$ can be packed without overlap. Constructions are known for a certain range of n, and summin...
Ausführliche Beschreibung

Abstract More than half a century ago Martin Gardner popularized a question leading to the benchmark problem of determining the minimum side length of a square into which the squares of sizes %$1,2,\dots ,n%$ can be packed without overlap. Constructions are known for a certain range of n, and summing up the areas yields that a packing in a square of size smaller than %$N:= \!\sqrt{n(n+1)(2n+1)/6)} %$ is not possible. Here we prove that an asymptotically minimal packing exists in a square of size %$N+cn+O(\!\sqrt{n})%$ with %$c<1%$, and such a packing is achievable with guillotine-cuts. An improved construction is also given for the case where the constraint of guillotine cutting is dropped. Ausführliche Beschreibung