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, Jeremy Tan, and Jonathan Viquerat.
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)
| 3.129+
(JV)
| 3.304+
| 3.613+
| 3.834+
(MM)
|
|---|
| 2
| 2.460+
(MM)
| (5 + 9√17)/16
(MM)
| (16 + 32√2)/21
(MM)
| 3.097+
(MM)
| 3.232+
(JV)
| 3.395+
(MM)
| 3.676+
(MM)
| 3.853+
(MM)
| 3.979+
(JV)
|
|---|
| 3
| 2.731+
(MM)
| 3.040+
(MM)
| 3.152+
(JV)
| 3.234+
(MM)
| 3.467+
(MM)
| 2 + √3
(MM)
| 3.892+
(JV)
| 4.014+
(JV)
| 1 + 2√21/3
(MM)
|
|---|
| 4
| √10
(MM)
| (6 + 5√42)/12
(MM)
| 3.325+
(JV)
| 3.529+
(MM)
| 3.782+
(MM)
| 3.927+
(JV)
| 4.036+
(JV)
| 4.095+
(JV)
| 4.269+
(JV)
|
|---|
| 5
| 3.268+
(MM)
| (3 + 9√33)/16
(MM)
| 3.623+
(MM)
| 1 + 2√2
(MM)
| 3.968+
(JV)
| 4.075+
(JV)
| 4.154+
(MM)
| 4.284+
(JV)
| 4.376+
(JV)
|
|---|
| 6
| 3.513+
(MM)
| (3 + 5√15)/6
(MM)
| 3.917+
(MM)
| 4.023+
(MM)
| 4.130+
(MM)
| 4.219+
(JV)
| 4.317+
(JV)
| 4.392+
(MM)
| 4.619+
(JV)
|
|---|
| 7
| 3.844+
(MM)
| 3.996+
(JV)
| 65/16
(MM)
| 4.177+
(JV)
| 17/4
(MM)
| 4.351+
(JV)
| 4.474+
(JV)
| 4.661+
(JV)
| 4.760+
(JV)
|
|---|
| 8
| 4.057+
(MM)
| √17
(MM)
| 4.218+
(JV)
| 4.303+
(JV)
| 4.396+
(JV)
| 4.499+
(JV)
| 4.686+
(JV)
| 4.795+
(JV)
| 4.956+
(JV)
|
|---|
| 9
| √1105/8
(MM)
| 4.276+
(JV)
| 4.354+
(JV)
| 4.443+
(JV)
| 4.586+
(JV)
| 4.737+
(JV)
| 4.858+
(JV)
| 5.013+
(JV)
| 5.088+
(JV)
|
|---|
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)
| 4.705+
(JV)
| 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)
| 4.254+
(JV)
| 4.606+
(JV)
| 4.729+
(MM)
| 4.953+
(MM)
|
|---|
| 4
| 2.913+
(JV)
| 3.152+
(JV)
| 3.429+
(MM)
| 2 + √3
(JD)
| 4.084+
(JV)
| 4.304+
(MM)
| 4.660+
(JV)
| 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)
| 4.718+
(JV)
| 4.948+
(MM)
| 5.137+
(MM)
|
|---|
| 6
| 3.113+
(JV)
| (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
| 3.154+
(JV)
| (7 + 2√3)/3
(MM)
| 2 + √3
(MM)
| 4.152+
(JV)
| 4.418+
(JV)
| (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)
| 4.027+
(JV)
| 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)
| 3.958+
(JV)
| 4.134+
(JV)
| (7 + √3)/2
(MM)
| 4.601+
(JV)
| 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)
| 2.856+
(JV)
| 3
(MM)
| 3.272+
(MM)
| 3.476+
(JV)
| 3.654+
(JV)
|
|---|
| 3
| 2
(MM)
| 2.206+
(MM)
| (21 + 5√3)/12
(MM)
| 2.718+
(JV)
| 2.959+
(MM)
| 3.096+
(MM)
| 3.362+
(JV)
| 3.557+
(JV)
| 3.739+
(JV)
|
|---|
| 4
| 2.062+
(MM)
| (3 + √3)/2
(MM)
| 2.615+
(MM)
| 2.803+
(JV)
| 3
(MM)
| 3.238+
(MM)
| 3.445+
(JV)
| 3.614+
(JV)
| 3.795+
(JV)
|
|---|
| 5
| 2.231+
(MM)
| 2.569+
(MM)
| 2.687+
(JV)
| 2.917+
(JV)
| 3.130+
(JV)
| 3.345+
(JV)
| 3.522+
(JV)
| 3.696+
(JV)
| 3.872+
(JV)
|
|---|
| 6
| 2.462+
(MM)
| 2.601+
(JV)
| 2.862+
(JV)
| 3.026+
(JV)
| 3.238+
(JV)
| 3.434+
(JV)
| 3.600+
(JV)
| 3.755+
(JV)
| 3.915+
(JV)
|
|---|
| 7
| 2.565+
(JV)
| 2.797+
(MM)
| 2.953+
(JV)
| 3.130+
(JV)
| 3.335+
(JV)
| 3.516+
(JV)
| 3.682+
(JV)
| 3.850+
(JV)
| 3.994+
(JV)
|
|---|
| 8
| 2.718+
(JV)
| 2.869+
(JV)
| 3.084+
(JV)
| 3.246+
(JV)
| 3.426+
(JV)
| 3.612+
(JV)
| 3.743+
(JV)
| 3.883+
(JV)
| 4.034+
(JV)
|
|---|
| 9
| 2.796+
(JV)
| 3.035+
(JV)
| 3.167+
(JV)
| 3.342+
(JV)
| 3.523+
(JV)
| 3.677+
(JV)
| 3.825+
(JV)
| 4.003+
(JV)
| 4.135+
(JV)
|
|---|
If you can extend any of these results, please
e-mail me.
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