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?
ANSWERS
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) → 1 as n → ∞.
Here are the best known solutions, together with the proportion of the area covered. Click on the pictures for the figures of n regions in squares of area n.
n=1
m=2
2√2 – 2 = .828+
| m=3
(2√3 + 3) / 8 = .808+
(Gavin Theobald)
| m=4
1
| m=5
.889+
|
m=6
7/√6 – 2 = .857+
(Gavin Theobald)
| m=7
(25√7 – 49) / 20 = .857+
(Gavin Theobald)
| m=8
√2 – 1/2 = .914+
| m=9
1
|
m=10
.906+
(Gavin Theobald)
| m=11
2√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=16
1
| |
n=2
m=3
2√3 – 5/2 = .964+
| m=4
1
| m=5
.948+
(Gavin Theobald)
| m=6
(51 - 26√3)/6 = .994+
(Gavin Theobald)
|
m=7
.969+
(Gavin Theobald)
| m=8
1
| 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=16
1
| |
n=3
m=4
2√3 – 5/2 = .964+
| m=5
.960+
(Gavin Theobald)
| m=6
6√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=12
1
| 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=7
63/64 = .984+
(Joe DeVincentis)
| m=8
1
|
m=9
1
| 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=16
1
| |
n=5
m=6
.973+
(Maurizio Morandi)
| m=7
35/36 = .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=14
35/36 = .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.
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