# Problem of the Month (October 2014)

If n unit squares are placed into a square of area n, what is the smallest possible maximum area of overlap of two of the squares? What if the n squares are placed in a square of integer side m, with m2 ≤ n ? More generally, how do you place unit squares in the plane with density δ with the smallest possible overlap?

Below are the placements of squares with the smallest known overlaps:

n Squares in a Square of Area n
 n=10 n=26 – 4√2.343+ n=321 – 12 √3.215+ n=40 n=5.110+(M. Morandi) n=6.132+(M. Morandi) n=7.127+(M. Morandi)
 n=8.070+ n=90 n=10.092+(Maurizio Morandi) n=11.087+(Maurizio Morandi)
 n=12.085+(Maurizio Morandi) n=13.081+(Maurizio Morandi) n=14.066+(Maurizio Morandi) n=15.037+
 n=160 n=17.070+(Joe DeVincentis) n=18.063+(Joe DeVincentis)
 n=19.073+(Maurizio Morandi) n=20.066+(Joe DeVincentis) n=21.070+(Maurizio Morandi)
 n=22.053+(Maurizio Morandi) n=23.041+(Joe DeVincentis) n=24.023+(Joe DeVincentis)
 n=250 n=26.062+(Joe DeVincentis) n=27.059+(Joe DeVincentis)

n Squares in a Square of Side 2
 n=40 n=51/4 n=61/3 n=7√2 – 1(M. Morandi) n=8(7–√13)/8 n=9(3+2√2)/12(M. Morandi) n=101/2(M. Morandi) n=112–√34/4(M. Morandi) n=1258–10√33(M. Morandi) n=13(√17–3)/2(M. Morandi) n=153/5(M. Morandi) n=1816/25(M. Morandi) n=212/3(M. Morandi) n=2525/36(J. DeVincentis) n=285/7(M. Morandi) n=3236/49(M. Morandi) n=363/4(M. Morandi) n=4149/64(J. DeVincentis) n=457/9(M. Morandi) n=5064/81(M. Morandi)

n Squares in a Square of Side 3
 n=90 n=10.143+(Maurizio Morandi) n=11.183+(Maurizio Morandi) n=12.197+(Joe DeVincentis) n=13√5 – 2(Bryce Herdt) n=14.274+(Joe DeVincentis) n=15.289+(Maurizio Morandi) n=16(21–√249)/16(Maurizio Morandi) n=181/3(Maurizio Morandi) n=212/5(Maurizio Morandi) n=22√2–1(Maurizio Morandi) n=243/7(Maurizio Morandi) n=254/9 n=2810/21(Maurizio Morandi) n=321/2(Joe DeVincentis) n=33(√13–2)/3(Joe DeVincentis) n=3615/28(Maurizio Morandi) n=405/9(Maurizio Morandi) n=419/16(Joe DeVincentis) n=457/12(Maurizio Morandi) n=503/5(Joe DeVincentis)

n Squares in a Square of Side 4
 n=160 n=17.092+(Maurizio Morandi) n=18.104+(Bryce Herdt) n=19.133+(Joe DeVincentis) n=20.139+(Joe DeVincentis) n=21.179+(Maurizio Morandi) n=22(√33–5)/4(Maurizio Morandi) n=233/14(Joe DeVincentis) n=24.221+(Joe DeVincentis) n=25(11–√73)/10(Maurizio Morandi) n=271/4(Maurizio Morandi) n=2810/37(Maurizio Morandi) n=304/13(Maurizio Morandi) n=325/16(Maurizio Morandi) n=351/3(Maurizio Morandi) n=365/14(Maurizio Morandi) n=408/21(Maurizio Morandi) n=4125/64(Maurizio Morandi) n=442/5(Maurizio Morandi) n=455/12(Maurizio Morandi) n=46.434+(Joe DeVincentis) n=507/16(Joe DeVincentis)

n Squares in a Square of Side 5
 n=250 n=26.076+(Joe DeVincentis) n=273/2–√2(Maurizio Morandi) n=28.093+(Maurizio Morandi) n=29.107+(Joe DeVincentis) n=301/9(Joe DeVincentis) n=32(19–6√2)/72(Maurizio Morandi) n=33.151+(Joe DeVincentis) n=34.175+(Joe DeVincentis) n=355/27(Maurizio Morandi) n=37(47–√1409)/48(Maurizio Morandi) n=391/5(Maurizio Morandi) n=405/21(Maurizio Morandi) n=421/4(Maurizio Morandi) n=449/35(Maurizio Morandi) n=483/11(Maurizio Morandi) n=503/10(Maurizio Morandi)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/30/14.