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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 ? For example, when n=1 and m is small, here are the best known results:

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 .854+ (Gavin Theobald)
m=8 √2 – 1/2 = .914+	m=9 1	m=10 .906+ (Gavin Theobald)

Can you improve any of these results? What are the best results you can find for larger n ? Does the unfilled space approach 0 as n and m approach ∞ ?

You can see all the best known results here.

Submit your answers here.

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