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#179 Martian food

It is the year 2100 and the Mars pioneers have built an agri-bubble in which they will be able to cultivate their own food. The crop is a form of grass that grows at a steady rate and can be harvested and turned into nutritious protein snacks (yum!). Now it is time to populate the planet.

¿ìè¶ÌÊÓƵs have figured out that if there are 40 adults living in the bubble, the crop will only feed them for 20 days. However, with only 20 adults, the crop will keep them going for 60 days – so half as many adults can survive for three times as long! Why? Because without overharvesting, the crop is able to replenish itself.

Of course, the pioneers want a food supply that keeps the population sustained indefinitely. Based on the numbers above, how many people should be in the first Mars cohort?

#178 Hydra

Solution

The monster is completely tame when all the heads show prime numbers. We can factorise 418 as 2×11×19 and 651 as 3×7×31. This leaves 113 – (2 + 11 + 19 + 3 + 7 + 31), which is 40, so the remaining heads must be prime numbers not already used that sum to 40. The only way to do this is as 17+23, so the mystery third head must have shown 17×23 = 391.