Problem of the Month (November 2016)

What is the largest area that can be filled in a regular hexagon by m regular hexagons of one size and n regular hexagons of another size if none of the hexagons overlap?

ANSWERS

George Sicherman, Maurizio Morandi, and Joe DeVincentis sent improvements.

Here are the best known answers:

m \ n2345678
0
1/2 = .500
3/4 = .750
36/49 = .734+
45/64 = .703+
18/25 = .720
(Maurizio Morandi)
21/25 = .840
(Maurizio Morandi)
432/(446+85√3) = .728+
(Maurizio Morandi)
2
25/32 = .781+
(George Sicherman)
7/8 = .875
38/49 = .775+
21/25 = .840
(Maurizio Morandi)
7/8 = .875+
171/200 = .855
(Maurizio Morandi)
144/169 = .852+
(George Sicherman)
3
15/16 = .937+
31/36 = .861+
8/9 = .888+
11/12 = .916+
.862+
(Maurizio Morandi)
.878+
(Maurizio Morandi)
4
40/49 = .816+
21/25 = .840
(George Sicherman)
42/49 = .857+
8/9 = .888+
44/49 = .897+
5
290/361 = .803+
(George Sicherman)
1007/1225 = .822+
(Maurizio Morandi)
11/12 = .916+
53/64 = .828+
(George Sicherman)
6
5/6 = .833+
17/18 = .944+
8/9 = .888+
7
546/625 = .873+
(Maurizio Morandi)
549/625 = .878+
(George Sicherman)
8
712/841 = .846+
(Maurizio Morandi)

m \ n9101112131415
0
729/(24+5√3)2 = .683+
(Maurizio Morandi)
70–40√3 = .717+
(Maurizio Morandi)
66(2–√3)/25 = .707+
(Maurizio Morandi)
12/16 = .750
13/16 = .812+
1050/1369 = .766+
(Maurizio Morandi)
13500/(73+35√3)2 = .756+
(Maurizio Morandi)
2
.819+
(Maurizio Morandi)
.835+
(Maurizio Morandi)
1339/1600 = .836+
(Maurizio Morandi)
.858+
(Maurizio Morandi)
.879+
(Maurizio Morandi)
.844+
(Maurizio Morandi)
.853+
(Maurizio Morandi)
3
.894+
(Maurizio Morandi)
29/32 = .906+
59/64 = .921+
15/16 = .937+
.882+
(Maurizio Morandi)
.892
(Maurizio Morandi)
.902
(Maurizio Morandi)
4
45/49 = .918+
.850+
(Maurizio Morandi)
.862+
(Maurizio Morandi)
.873+
(Maurizio Morandi)
.885+
(Maurizio Morandi)
.866+
(Maurizio Morandi)
57/64 = .890+
(Maurizio Morandi)
5
27/32 = .843+
(George Sicherman)
55/64 = .859+
(George Sicherman)
7/8 = .875
(George Sicherman)
57/64 = .890+
(George Sicherman)
57/64 = .890+
57/64 = .890+
(Maurizio Morandi)
108/125 = .864
(Maurizio Morandi)
6
11/12 = .916+
16560/18769 = .882+
(Maurizio Morandi)
315/361 = .872+
(George Sicherman)
22/25 = .880
(George Sicherman)
29/32 = .906+
85/96 = .885+
(Maurizio Morandi)
173/192 = .901+
(George Sicherman)
7
8/9 = .888+
73/81 = .901+
74/81 = .913+
25/27 = .925+
2217/2500 = .886+
(Maurizio Morandi)
1113/1250 = .890+
(Maurizio Morandi)
351/392 = .895+
(Maurizio Morandi)
8
929/1089 = .853+
(Maurizio Morandi)
6/7 = .857+
57/64 = .890+
(Maurizio Morandi)
44/49 = .897+
(Maurizio Morandi)
357/400 = .892+
43/50 = .860
(George Sicherman)
87/100 = .870
(George Sicherman)
9
106/121 = .876+
57/64 = .890+
(Maurizio Morandi)
141/169 = .834+
(George Sicherman)
57/64 = .890+
361/400 = .902+
27/32 = .843+
(Joe DeVincentis)
3555/4096 = .867+
(Joe DeVincentis)
10
285/338 = .843+
(Joe DeVincentis)
27/32 = .843+
(Joe DeVincentis)
29/32 = .906+
73/80 = .912+
146/169 = .863+
(George Sicherman)
70/81 = .864+
(Joe DeVincentis)
11
55/64 = .859+
(George Sicherman)
59/64 = .921+
(Joe DeVincentis)
369/400 = .922+
29/32 = .906+
59/64 = .921+
(Maurizio Morandi)
12
15/16 = .937
(George Sicherman)
373/400 = .932+
89/100 = .890
(Joe DeVincentis)
9/10 = .900
(George Sicherman)
13
24128/26569 = .908+
(Maurizio Morandi)
131/144 = .909+
(Joe DeVincentis)
11/12 = .916+
(George Sicherman)
14
70/81 = .864+
(George Sicherman)
71/81 = .876+
(Joe DeVincentis)
15
1160/1323 = .876+
(Maurizio Morandi)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/23/16.