# Problem of the Month (May 2008)

Let n < m be positive integers. What is the largest shape with the property that n of them can be packed inside a square of area n, and m of them can be packed inside a square of area m ? Can you beat any of the results below?

Jeremy Galvagni noticed that if m=(a/b)2n, then area 1 could be trivially covered. He also showed that when n=m-k, that area √(1-k/m) can be covered using thin rectangles:

He later improved this bound to area 2√(m/n) + 3√(n/m) – 4 when n is even, and √(n/m) + 2n√(n/m)/(n+1) + 2√(mn)/(n+1) – 4n/(n+1) when n is odd.

Joe DeVincentis noticed that if k is the smallest square number greater than n, then we can use n of k squares, for a fill fraction of n/k > n/(n+2√n), which approaches 1 as n approaches ∞.

Here are the best known solutions, together with the proportion of the area covered. Click on the pictures for the figures of m regions in squares of area m.

n=1
 m=22√2 – 2 = .828+ m=3(2√3 + 3) / 8 = .808+(Gavin Theobald) m=41 m=5.889+ m=67/√6 – 2 = .857+(Gavin Theobald) m=7(25√7 – 49) / 20 = .857+(Gavin Theobald) m=8√2 – 1/2 = .914+ m=91 m=10.906+(Gavin Theobald) m=112√11 – 23/4 = .883+(Gavin Theobald) m=12.897+(Gavin Theobald) m=13.892+(Gavin Theobald) m=14.913+(Gavin Theobald) m=15.937+(Gavin Theobald) m=161

n=2
 m=32√3 – 5/2 = .964+ m=41 m=5.948+(Gavin Theobald) m=6(51 - 26√3)/6 = .994+(Gavin Theobald) m=7.969+(Gavin Theobald) m=81 m=9.973+(Gavin Theobald) m=10√10 + √5 - √2 - 3 = .984+(Maurizio Morandi) m=11.957+(Gavin Theobald) m=12(36 - 5√6)/24 = 0.989+(Maurizio Morandi) m=13.963+(Gavin Theobald) m=14.986+(Maurizio Morandi) m=15.978+(Gavin Theobald) m=161

n=3
 m=42√3 – 5/2 = .964+ m=5.960+(Gavin Theobald) m=66√2 – 15/2 = .985+(Gavin Theobald) m=7(224√2 – 35) / 289 = .975+(Gavin Theobald) m=8(2√6 – 2) / 3 = .966+ m=9.974+(Gavin Theobald) m=10.975+(Maurizio Morandi) m=11.961+(Maurizio Morandi) m=121 m=13.964+(Maurizio Morandi) m=14(224√2 – 35) / 289 = .975+(Gavin Theobald) m=15.966+(Gavin Theobald) m=16√3 – 3/4 = .982+(Gavin Theobald)

n=4
 m=5.974+(Maurizio Morandi) m=6(36 – 5√6) / 24 = .989+(Maurizio Morandi) m=763/64 = .984+(Joe DeVincentis) m=81 m=91 m=10.986+(Maurizio Morandi) m=11.981+(Maurizio Morandi) m=12(36 - 7√3)/24 = .994+(Maurizio Morandi) m=13.985+(Maurizio Morandi) m=14.986+(Maurizio Morandi) m=15.978+(Gavin Theobald) m=161

n=5
 m=6.973+(Maurizio Morandi) m=735/36 = .972+ m=8.975+(Maurizio Morandi) m=9.977+(Maurizio Morandi) m=10.974+(Maurizio Morandi) m=11.9618+(Maurizio Morandi) m=12.965+(Maurizio Morandi) m=13.964+(Gavin Theobald) m=1435/36 = .972+ m=15.954+(Gavin Theobald) m=16(4√5 – 4)/5 = .988+(Gavin Theobald)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/16/09.