surely the way to answer this query is to say 'what does each manifestation of the space-craft need to carry at each point and for what purpose?' This is almost surely best calculated in reverse from end to start.

Simplifying (and probably getting wrong) the many actual "burn" and "separation" stages (including potential abort burn stages) the Saturn 5 rocket had to lift off the ground...

1) fuel to allow any deceleration burn of the final earth re-entry module from velocity at the end of the (gravity influenced) transit from moon to the velocity necessary to renter earth's atmosphere
+2) fuel to accelerate the earth re-entry module (including fuel#1) from the lunar orbit to decent velocity to make trip back to earth
+3) fuel to lift (top section) of lunar module (2 crew) from moon surface to orbit
+4) fuel to decelerate and control lunar lander (including fuel#3) during landing on moon
+5) fuel to decelerate lunar lander (including fuels#3-4) from orbit into decent*
+6) fuel to decelerate various lunar module elements (including fuels#1-5) from velocity when reaching moon to lunar orbital velocity
+7) fuel to accelerate all modules travelling to moon (including fuel#1-6) from near earth orbit velocity (assuming such an orbit was involved - not sure) to velocity required to reach moon in reasonable time (subject to fluctuations in speed caused by gravitational fields)
+8) fuel to lift all successive main rocket stages (including the various lunar-travelling vehicles plus fuel#1-7) from the earths surface to near earth orbit (minus weight of successive rocket stages after the point where dropping away).

Contrasting (8), which is requirement for Saturn 5, with (5), which is the requirement for the lunar landing vehicle (albeit before it leaves part of itself behind on the moon), there is clearly a huge discrepancy caused in part by the compound nature of the fuel requirements. Also surely the relative arithmetical relationship between the "strengths" of the two gravities (? 1/6) is compounded by the fact that any burn required to reach/leave near-orbit has to be somewhat longer (as well as stronger) for earth.

Clearly these cumulative / exponential effects are what the editors of Mensa would have been hoping to get members to recognise and comment upon in their responses. They are reminiscent of those sort of "how many trips does it take to ferry a fox, a chimpanzee and a water buffalo across a desert with a bicycle built for three?" sort of problems which you used to get in puzzle books for kids.

*Possibly the lunar lander module could have been decelerated (to start "falling" from orbit) by "burns" carried out by the orbiting module whilst they were still linked together (meaning the lander itself would not have had to be "big enough" to carry that extra bit of deceleration fuel itself, but meaning that the orbiting module would have had to burn again after separation to stop itself from falling too and restore its orbit). This is probably overcomplicated and almost certainly less fuel efficient overall, but I can't figure that out as its a trade-off between the extra fuel (and capacity) used by the orbiter slowing and speeding itself (unnecessarily) against the fuel required for the lander to carry that extra "deceleration fuel"-capacity down to the surface. I suspect that the latter is probably marginal and still the most efficient overall (especially since it then leaves behind it's descent gear). Either way the relevant 'capacity' has to be conveyed to the moon (with the fuel implications that has). If the lander does its own deceleration that capacity also has to be lowered in a controlled way to the surface, but if the orbiter does the deceleration then that bit of capacity may also have to be brought back to earth orbit. If the deceleration burn is short but uses lots of fuel then it may work out cheaper to make the orbiter do it. Unfortunately I think it takes a rocket scientist to work that out...