2016/06/12 by Bentz, Wolfram
#05B40 #52C15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1606.03746
Let s(n) be the side length of the smallest square into which n non-overlapping unit squares can be packed. In 2010, the author showed that s(13)=4 and s(46)=7. Together with the result s(6)=3 by Keaney and Shiu, these results strongly suggest that s(m2-3)=m for m≥ 3, in particular for the values m=5,6, which correspond to cases that lie in between the previous results. In this article we show that indeed s(m2-3)=m for m=5,6, implying that the most efficient packings of 22 and 33 squares are the trivial ones. To achieve our results, we modify the well-known method of sets of unavoidable points by replacing them with continuously varying families of such sets.