# Problem of the Month (April 2015)

What is the largest non-overlapping area that can be filled in a regular hexagon or an equilateral triangle by m congruent regular hexagons and n congruent equilateral triangles?

Solutions this month were received from David Cantrell, Maurizio Morandi, Jeremy Galvagni, Joe DeVincentis, and George Sicherman.

Here are the best known answers:

Best Packings of Hexagons
m \ n023456789
0
0 = .000

1/2 = .500

3/4 = .750

36/49 = .734+

45/64 = .703+

18/25 = .720 (MM)

21/25 = .840 (MM)

.728+ (MM)

.683+ (MM)
1
1/2 = .500

2/3 = .666+

19/24 = .791+

75/98 = .765+

143/192 = .744+ (GS)

5/6 = .833+

127/150 = .846+ (MM)

61/78 = .782+ (MM)

59/72 = .819+
2
4/9 = .444+

5/6 = .833+

5/6 = .833+

39/49 = .795+

3/4 = .750 (GS)

21/25 = .840 (MM)

64/75 = .853+ (MM)

5/6 = .833+

.799+ (MM)
3
.510+

41/50 = .820

7/8 = .875

81/98 = .826+

77/90 = .855+

.835+ (MM)

131/150 = .873+ (MM)

617/722 = .854+ (MM)

43/50 = .860
4
2/3 = .666+

8/9 = .888+

11/12 = .916+

11/12 = .916+

.870+ (MM)

68/75 = .906+

326/363 = .898+ (MM)

68/75 = .906+ (MM)

281/300 = .936+ (MM)
5
5/6 = .833+

137/150 = .913+

23/24 = .958+

17/18 = .944+

.940+ (MM)

.959+ (MM)

181/192 = .942+

3557/3750 = .948+

143/150 = .953+ (JD)
7
.715+ (DC)

.802+ (MM)

.916+ (MM)

41/48 = .854+ (GS)

111/128 = .867+

717/800 = .896+ (MM)

49/54 = .907+

107/120 = .891+ (GS)

897/1024 = .875+ (MM)
8
.725+ (DC)

5/6 = .833+

25/27 = .925+

43/48 = .895+ (GS)

57/64 = .890+

102/121 = .842+ (GS)

25/27 = .925+

8422/9747 = .864+ (MM)

43/48 = .895+
9
.728+

8/9 = .888+

11/12 = .916+

22/25 = .880 (GS)

117/128 = .914+

314/363 = .865+ (MM)

17/18 = .944+

8/9 = .888+ (GS)

9/10 = .900 (GS)

Best Packings of Triangles
m \ n0123456789
0
0 = .000

2/3 = .666+

12/25 = .480

18/25 = .720

2/3 = .666+

.643+ (MM)

.754+ (MM)

21/32 = .656+

432/625 = .691+

18/25 = .720 (JD)
2
1/2 = .500

8/9 = .888+

5/6 = .833+

4/5 = .800

7/8 = .875

.805+ (MM)

43/50 = .860

205/242 = .847+ (MM)

43/50 = .860 (MM)

181/200 = .905 (MM)
3
3/4 = .750

1 = 1.000

87/100 = .870

93/100 = .930

11/12 = .916+

.910+ (MM)

.938+ (MM)

117/128 = .914+

2307/2500 = .922+

93/100 = .930 (JD)
5
5(1–√3/2) = .669+

23/27 = .851+

8/9 = .888+

23/25 = .920

23/27 = .851+

165/196 = .841+ (GS)

.890+ (MM)

12941/15129=.855+(MM)

7721/8649 = .892+ (MM)

197/225 = .875+ (JD)
6
.676+ (MM)

5/6 = .833+

22/25 = .880

24/25 = .960

314/363 = .865+

122/135 = .903+

.890+ (MM)

206/225 = .915+ (MM)

326/361 = .903+ (MM)

68/75 = .906+
7
7/9 = .777+

17/18 = .944+

25/27 = .925+

1 = 1.000

17/18 = .944+

.913+ (MM)

211/225 = .937+

1015/1089 = .932+ (MM)

211/225 = .937+ (MM)

431/450 = .957+ (MM)
8
8/9 = .888+

26/27 = .962+

212/225 = .942+

218/225 = .968+

26/27 = .962+

.960+ (MM)

.972+ (MM)

277/288 = .961+

5432/5625 = .965+

218/225 = .968+ (JD)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/1/15.