Guillotine cutting is asymptotically optimal for packing consecutive squares

Abstract

AbstractMore 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 $$1,2,\dots ,n$$ 1 , 2 , ⋯ , 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 $$N:= \!\sqrt{n(n+1)(2n+1)/6)} $$ N : = n ( n + 1 ) ( 2 n + 1 ) / 6 ) is not possible. Here we prove that an asymptotically minimal packing exists in a square of size $$N+cn+O(\!\sqrt{n})$$ N + c n + O ( n ) with $$c<1$$ 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.