Problem of the Month (March 2010)

Packing circles, squares, or triangles inside each other has been well-studied. See my Packing Center for details. This month we investigate packing two of these shapes (with sides or diameter 1) inside one of them. Can you fill in the missing entries, or improve the existing entries, in the tables below?

ANSWERS

This month's solvers include Joe DeVincentis, Maurizio Morandi, George Sicherman, Dan Dima, Jeremy Galvagni, Károly Hajba, and Jeremy Tan.

Squares and Triangles in Squares

m/n	1	2	3	4	5	6	7	8	9
1	1 + √6/4	1.762+ (MM)	1 + √3/2	1 + √3/2	2 (MM)	2.173+ (MM)	4 - √3 (MM)	2.398+ (MM)	2.490+ (MM)
2	2	2	2	2.171+ (MM)	2.329+ (MM)	6√3 - 8 (MM)	2.523+ (MM)	8/√3 - 2 (MM)	1 + √3 (MM)
3	2	2.177+	(3 + √3)/2 (MM)	2.452+ (MM)	2.595+ (MM)	(11√3 - 3)/6 (MM)	2.767+ (MM)	(181 + 17√3)/74 (MM)	2 + √3/2
4	2√3 - 1	(19+8√3)/13 (MM)	2.630+ (MM)	1 + √3 (MM)	(35 + √3)/13 (MM)	2 + √3/2	2 + √3/2	2 + √3/2	2.980+ (MM)
5	(77+15√3)/37 (MM)	(5√3 - 3)/2 (MM)	2 + √3/2	2 + √3/2	(28 + 6√3)/13 (MM)	3	3	(3√3 + 1)/2	3/2 + √3 (MM)
6	3	3	3	3	3	3.132+ (MM)	7 - 13√3/6 (MM)	5√3/2 - 1 (MM)	4√3 - 7/2 (MM)
7	3	3	3	3.149+ (MM)	(22 + 12√3)/13 (MM)	(5 + √3)/2 (MM)	2√3 (MM)	3.521+ (MM)	1 + 3√3/2 (MM)
8	3	2 + 2/√3 (MM)	(38 + 3√3)/13 (MM)	(5 + √3)/2 (MM)	2√3 (MM)	3 + 1/√3 (MM)	3.624+ (MM)	(3 + 11√3)/6 (MM)	2 + √3 (MM)
9	2√3	2√3	(32 + 8√3)/13 (MM)	3 + 1/√3	(9 - √3)/2 (MM)	2 + √3 (MM)	(156 + 13√3)/47 (MM)	23√3 - 36 (MM)	3 + √3/2

Squares and Triangles in Triangles

m/n	1	2	3	4	5	6	7	8	9
1	1 + 2/√3	1 + √2	2 + 1/√3	5/2 + 1/√3 (MM)	3.107+ (MM)	2 + 2/√3	(11 + √5)/4 (MM)	3.574+ (MM)	3 + 1/√3 (MM)
2	2 + 2/√3	2 + 2/√3	2 + 2/√3	2 + 2/√3	(24 + 4√3)/9 (MM)	3 + 1/√3 (MM)	3.696+ (MM)	3 + √3/2 (MM)	4.144+ (MM)
3	1 + 4/√3	2 + 5√3/6 (MM)	3.572+ (MM)	2 + √3	3 + √3/2 (MM)	2 + √2 + 1/√3 (MM)	3 + 2/√3	3 + 2/√3	3 + 2/√3
4	3 + 2/√3	3 + 2/√3	3 + 2/√3	5/2 + √3 (MM)	5/2 + √3 (MM)	2 + 4/√3	2 + 4/√3 (MM)	3 + 6√3/7 (MM)	(42 + 16√3)/15 (MM)
5	2 + 4/√3	2 + 4/√3	2 + 4/√3	2 + 4/√3	3 + 5/2√3 (MM)	(24 + 10√3)/9 (MM)	3 + √3 (MM)	5/2 + 4/√3 (MM)	(45 + 8√3)/12 (MM)
6	1 + 2√3	1 + 2√3 (MM)	2 + 3√3/2 (MM)	(19 + 8√3)/7 (MM)	4.810+ (MM)	2 + 5/√3 (MM)	3 + 7√3/6 (JD)	5.110+ (MM)	(30 + 10√3)/9 (MM)
7	4 + 2/√3	4 + 2/√3	4 + 2/√3	4 + 2/√3 (MM)	4 + 2/√3 (MM)	4 + 2/√3 (MM)	3 + 4/√3	3 + 4/√3	3 + 4/√3
8	5.301+ (MM)	5.301+ (MM)	3 + 4/√3 (MM)	3 + 4/√3 (MM)	(15 + 10√3)/6 (JD)	4 + 5/2√3 (MM)	2 + 2√3 (MM)	(41 + 8√3)/10 (MM)	(18 + 6√3)/5 (MM)
9	3 + 4/√3	(15 + 10√3)/6 (MM)	2 + 2√3	2 + 2√3	2 + 2√3 (MM)	3 + 3√3/2 (MM)	(26 + 8√3)/7 (MM)	4 + 22√3/21 (MM)	3 + 5/√3 (MM)

Squares and Circles in Squares

m/n	1	2	3	4	5	6	7	8	9
1	(6 + √2)/4	2	2	1 + √2	2.683+	2 + √3/2	2.956+	3	3.373+
2	2	2	2.549+	2 + 1/√2	2 + √3/2	2 + (√2+√6)/4	3	3.376+ (MM)	3.511+
3	2	(6 + 3√2)/4	2.766+ (MM)	2.923+	3	3	3.376+ (MM)	3.511+	3.652+ (MM)
4	2 + 1/√2	(10 + √2)/4	(21 + √7)/8	3	3	3.376+ (MM)	3.547+ (MM)	3.652+ (MM)	3.788+ (MM)
5	2 + 2√2/3 (MM)	3	3	3	2 + √2 (JG)	3.567+ (MM)	3.683+ (MM)	19/5 (MM)	3 + √3/2
6	3	3	3	2 + √2 (JG)	(3 + 3√2)/2 (MM)	3 + 1/√2 (JD)	3.841+ (MM)	3.926+ (MM)	3.997+ (MM)
7	3	3	3.548+ (MM)	3.675+ (MM)	(4+6√2+√6)/4 (MM)	3.861+ (MM)	3.948+ (MM)	3.998+ (MM)	4
8	3	3.558+ (MM)	3 + 1/√2 (MM)	3.793+ (MM)	3.914+ (MM)	3+(√2+√6)/4 (MM)	4	4	4.349+ (MM)
9	3 + 1/√2	3 + 1/√2 (MM)	(14 + √2)/4 (JD)	3.922+ (MM)	3.995+ (MM)	4	4	4.349+ (MM)	4.488+ (MM)

Squares and Circles in Circles

m/n	1	2	3	4	5	6	7	8	9
1	17/8 (JD)	(2 + 3√6)/4 (MM)	2.523+ (MM)	2.806+ (MM)	3.064+ (MM)	3.134+ (MM)	3.304+	3.613+	3.834+ (MM)
2	2.460+ (MM)	(5 + 9√17)/16 (MM)	(16 + 32√2)/21 (MM)	3.097+ (MM)	3.233+ (MM)	3.395+ (MM)	3.676+ (MM)	3.853+ (MM)	4.002+ (MM)
3	2.731+ (MM)	3.040+ (MM)	√10 (MM)	3.234+ (MM)	3.467+ (MM)	2 + √3 (MM)	?	?	1 + 2√21/3 (MM)
4	√10 (MM)	(6 + 5√42)/12 (MM)	?	3.529+ (MM)	3.782+ (MM)	3.942+ (MM)	1 + 2√21/3 (MM)	4.125+ (MM)	?
5	3.268+ (MM)	(3 + 9√33)/16 (MM)	3.623+ (MM)	1 + 2√2 (MM)	?	?	4.154+ (MM)	?	?
6	3.513+ (MM)	(3 + 5√15)/6 (MM)	3.917+ (MM)	4.023+ (MM)	4.130+ (MM)	4.237+ (MM)	?	4.392+ (MM)	?
7	3.844+ (MM)	(2 + 3√22)/4 (MM)	65/16 (MM)	?	17/4 (MM)	?	?	?	?
8	4.057+ (MM)	√17 (MM)	?	√4745/16 (MM)	?	4.561+ (MM)	?	?	?
9	√1105/8 (MM)	√4745/16 (MM)	4.391+ (MM)	?	?	?	?	√26 (MM)	5.109+ (MM)

Triangles and Circles in Triangles

m/n	1	2	3	4	5	6	7	8	9
1	1 + 2/√3 (MM)	1 + √3	3.032+ (MM)	3.630+ (MM)	2 + √3	4.032+ (MM)	1 + 2√3 (MM)	4.706+ (MM)	3 + √3
2	1 + 2/√3 (JD)	1 + √3 (JD)	2 + 2/√3 (JD)	3.633+ (MM)	2 + √3 (JD)	4.120+ (MM)	1 + 2√3 (MM)	4.713+ (MM)	3 + √3 (JD)
3	(3 + √3)/2 (JD)	3.041+ (MM)	3.296+ (JD)	3.696+ (MM)	3.953+ (MM)	4.268+ (MM)	4.615+ (MM)	4.729+ (MM)	4.953+ (MM)
4	2.923+ (MM)	2 + 2/√3 (JD)	3.429+ (MM)	2 + √3 (JD)	4.087+ (JD)	4.304+ (MM)	4.702+ (MM)	3 + √3 (JD)	5.032+ (MM)
5	3 (JD)	2 + 2/√3 (MM)	3.548+ (MM)	(15 + 5√3)/6 (MM)	3 + 2/√3 (JD)	4.435+ (MM)	4.729+ (MM)	4.948+ (MM)	5.137+ (MM)
6	3.115+ (MM)	(5 + √3)/2 (MM)	2 + √3 (JD)	4.077+ (MM)	4.255+ (MM)	4.547+ (MM)	3 + √3 (JD)	5.032+ (MM)	(24 + 13√3)/9 (MM)
7	2 + 2/√3 (MM)	(7 + 2√3)/3 (MM)	2 + √3 (MM)	3 + 2/√3 (JD)	4.424+ (MM)	(7 + 4√3)/3 (MM)	(11 + 5√3)/4 (MM)	5.114+ (MM)	5.284+ (MM)
8	(5 + √3)/2 (MM)	5/2 + 2/√3 (MM)	4.032+ (MM)	3 + 2/√3 (JD)	4.496+ (MM)	3 + √3 (JD)	5.032+ (MM)	4 + 2/√3 (JD)	3 + 4/√3 (MM)
9	3 + 1/√3 (MM)	3.980+ (MM)	3 + 2/√3 (JD)	(7 + √3)/2 (MM)	4.608+ (MM)	4 + √3/2 (MM)	5.109+ (JD)	5.229+ (MM)	5.516+ (MM)

Triangles and Circles in Circles

m/n	1	2	3	4	5	6	7	8	9
1	1 + √3/2 (GS)	1/2 + √21/3 (GS)	2.248+ (MM)	2.528+ (MM)	2.827+ (MM)	3 (JD)	3.134+ (MM)	3.413+ (JT)	3.613+ (MM)
2	1.975+ (MM)	2.093+ (MM)	2.354+ (MM)	2.651+ (MM)	2.857+ (KH)	3 (MM)	3.272+ (MM)	?	?
3	2 (MM)	2.206+ (MM)	(21 + 5√3)/12 (MM)	2.781+ (JT)	2.959+ (MM)	3.096+ (MM)	3.389+ (MM)	?	?
4	2.062+ (MM)	(3 + √3)/2 (MM)	2.615+ (MM)	2.806+ (JT)	3 (MM)	3.238+ (MM)	?	?	?
5	2.231+ (MM)	2.569+ (MM)	?	?	?	?	?	?	?
6	2.462+ (MM)	2.654+ (JT)	?	3.057+ (MM)	?	?	?	?	3.923+ (MM)
7	2.617+ (JT)	2.797+ (MM)	3.035+ (JT)	3.272+ (JT)	3.400+ (JT)	?	?	?	?
8	2.733+ (JT)	2.912+ (MM)	3.152+ (MM)	3.337+ (JT)	?	3.644+ (JT)	?	3.909+ (JT)	?
9	2.797+ (JT)	3.063+ (MM)	?	3.387+ (MM)	?	?	?	?	4.332+ (JT)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/5/12.