A 450 mm (17.7 inch) diameter sphere containing a smaller sphere 284.4 mm (11.2 inches) equipped so that the small sphere carries 13.73 kg of LOX and the larger sphere contains 2.50 kg of LH2 and the sphere's themselves mass 1.05 kg. So we have a single sphere capable of a final velocity of;

Vf = 4.5 * ln(16.28 / 1.05) = 12.33 km/sec.

So, a take of weight for a sphere of 24.09 kg containing 15.23 kg of propellant (13.73 + 2.50) propels 8.86 kg through 4.5 km/sec. With 1.05 kg of 'structure' we have a structure fraction of 1.05 / 24.09 = 4.36% and propellant fraction of 15.23 / 24.09 = 63.23% leaving 100% - 63.23% - 4.36% = 32.41% payload.

Now, if we say a sphere carries a payload through a delta vee of 4.5 km/sec and we have four stages, two stages to orbit Earth, one stage to go from Earth orbit to the vicinity of the Moon, and finally, one stage to go from the Moon, to the Moon's surface, and back to Earth.

With a biosuit, and supplies for 14 days, an astronaut leaving Earth, massing 85 kg (on average) in a long-duration mechanical counter-pressure suit and supplies bringing the total to 140 kg. That's the payload.

Working backwards we have

140 kg / 0.3241 = 431.97

and so

431.97 * 0.0436 / 1.05 = 17.93 ~ 18

Eighteen spheres take a fully equipped astronaut with supplies for 12 days - which mean about 18 kilos of samples - from the lunar surface to Earth's surface.

We then do the same thing again using the 18 spheres + 140 kg as payload

18 * 16.28 + 140 = 433.04

and so;

433.04 / 0.3241 = 1,336.14

therefore;

1,336.14 * 0.0436 / 1.05 = 55.48 ~ 56

spheres... to take the return spheres and the astronaut from Earth orbit to the vicinity of the moon.

56 * 16.28 + 433.04 = 1,344.72

Using this as payload, we calculate the upper stage of the launcher... in terms of spheres.

1,344.72 / 0.3421 = 3930.79 kg

which translate to;

3930.79 * 0.0436 / 1.05 = 163.22 ~ 164 spheres

and so, to calculate the first stage and take off weight;

164 * 16.28 + 3930.79 = 6600.71

6600.71 * 0.0436 / 1.05 = 274.09 ~ 275

275 Stage 1
164 Stage 2
56 Stage 3
18 Stage 4

513 Total

513 * 16.28 + 140 = 8,491.64 kg TAKE OFF WEIGHT.

So, a disk of hexagonal close packed array of spheres 8.55 meters (28.0 ft) in diameter topped by another disk of hcp array of spheres 7.83 meters (25..7 ft) in diameter, is sufficient to take an astronaut to the moon and back, and return ALL the spheres to be reused to the launch point.