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Solutions for the smallest diameter circles into which $n$ unit-diameter circles can be packed have been proved optimal for $n$ = 1 through 10, based on a paper by R.L. Graham et.al. 'Dense packings of congruent circles in a circle', 1999, based on Kravitz, S. 'Packing Cylinders into Cylindrical Containers' Math. Mag. 40, 65-70, 1967.
| $1$ | $N_c<=1$ |
| $2$ | $N_c<=2$ |
| $1+0.6667\sqrt{3}$ | $N_c<=3$ |
| $1+\sqrt{2}$ | $N_c<=4$ |
| $1+\sqrt{2\left(1+\frac{1}{\sqrt{5}}\right)}$ | $N_c<=5$ |
| $1+\frac{1}{sin\left(\frac{\pi}{6}\right)}$ | $N_c<=7$ |
| $1+\frac{1}{sin\left(\frac{\pi}{7}\right)}$ | $N_c<=8$ |
| $1+\sqrt{2\left(2+\sqrt{2}\right)}$ | $N_c<=9$ |
| N | Factor | equation | density | optimal | Reference |
|---|---|---|---|---|---|
| 1 | 1.0 | 1.0 | trivially optimal | ||
| 2 | 2.0 | $1+1=1+1/sin(\pi/2)$ | 0.5 | trivially optimal | |
| 3 | 2.1547 | $1+2/\sqrt 3=1+1/sin(\pi/3)$ | 0.6466 | trivially optimal | |
| 4 | 2.4142 | $1+\sqrt 2=1+1/sin(\pi/4)$ | 0.6864 | trivially optimal | |
| 5 | 2.7013 | $1+\sqrt (2(1+1/\sqrt 5))=1+1/sin(\pi/5)$ | 0.6854 | proved optimal | Graham (1968) |
| 6 | 3.0 | $1+1/sin(\pi/6)$ | 0.6666 | proved optimal | Graham (1968) |
| 7 | 3.0 | $1+1/sin(\pi/6)$ | 0.7777 | trivially optimal | |
| 8 | 3.3048 | $1+1/sin(\pi/7)$ | 0.7328 | proved optimal | |
| 9 | 3.6131 | $1+\sqrt (2(2+\sqrt 2))=1+1/sin(\pi/8)$ | 0.6895 | proved optimal | Pirl (1969) |
| 10 | 3.813 | 0.6878 | proved optimal | Pirl (1969) | |
| 11 | 3.9238 | $1+1/sin(\pi/9)$ | 0.7148 | proved optimal | Melissen (1994) |
| 12 | 4.0296 | 0.7392 | proved optimal | Fodor (2000) | |
| 13 | 4.2361 | $2+\sqrt 5=1+1/sin(\pi/10)$ | 0.7245 | proved optimal | Fodor (2003) |
| 14 | 4.3284 | 0.7474 | conjectured optimal | ||
| 15 | 4.5214 | $1+\sqrt (6+2/\sqrt 2+4\sqrt (1+2/\sqrt 5))$ | 0.7339 | conjectured optimal | |
| 16 | 4.6154 | 0.7512 | conjectured optimal | ||
| 17 | 4.792 | 0.7403 | conjectured optimal | ||
| 18 | 4.8637 | $1+\sqrt 2 +\sqrt 6=1+1/sin(\pi/12)$ | 0.7611 | conjectured optimal | |
| 19 | 4.8637 | $1+\sqrt 2 +\sqrt 6=1+1/sin(\pi/12)$ | 0.8034 | proved optimal | Fodor (1999) |
| 20 | 5.1223 | 0.7623 | conjectured optimal | ||
| 21 | 5.2523 | 0.7612 | |||
| 22 | 5.3972 | 0.7435 | |||
| 23 | 5.5452 | 0.748 | |||
| 24 | 5.6517 | 0.7514 | |||
| 25 | 5.7528 | 0.7554 | |||
| 26 | 5.8282 | 0.7654 | |||
| 27 | 5.9064 | 0.774 | |||
| 28 | 6.0149 | 0.7739 | |||
| 29 | 6.1386 | 0.7696 | |||
| 30 | 6.1977 | 0.781 | |||
| 31 | 6.2915 | 0.7832 | |||
| 32 | 6.4294 | 0.7741 | |||
| 33 | 6.4867 | 0.7843 | |||
| 34 | 6.611 | 0.778 | |||
| 35 | 6.6972 | 0.7803 | |||
| 36 | 6.6747 | 0.7909 | |||
| 37 | 6.7588 | 0.81 | |||
| 38 | 6.9619 | 0.784 | |||
| 39 | 7.0579 | 0.7829 | |||
| 40 | 7.1238 | 0.7882 | |||
| 41 | 7.26 | 0.7779 | |||
| 42 | 7.3468 | 0.7781 | |||
| 43 | 7.4199 | 0.781 | |||
| 44 | 7.498 | 0.7826 | |||
| 45 | 7.5729 | 0.7847 | |||
| 46 | 7.6502 | 0.786 | |||
| 47 | 7.7242 | 0.7878 | |||
| 48 | 7.7913 | 0.7907 | |||
| 49 | 7.8869 | 0.7877 | |||
| 50 | 7.9475 | 0.7916 | |||
| 51 | 8.0275 | 0.7914 | |||
| 52 | 8.0847 | 0.7956 | |||
| 53 | 8.1796 | 0.7922 | |||
| 54 | 8.204 | 0.8023 | |||
| 55 | 8.2111 | 0.8158 | |||
| 56 | 8.3835 | 0.7968 | |||
| 57 | 8.4472 | 0.7988 | |||
| 58 | 8.5246 | 0.7982 | |||
| 59 | 8.5925 | 0.7991 | |||
| 60 | 8.6462 | 0.806 |