Ken's Puzzle of the Week

Slow Cows

Freddie the farmer has a paddock that he uses to graze his stock of cows. Each cow eats the same amount of grass each day, regardless of how many other cows are in the paddock and irrespective of the amount of grass left in the paddock. In an experiment, Freddie puts 6 cows into the paddock and he finds out it takes 3 days for them to eat all the grass. These 6 cows are then taken out of the paddock to allow the grass to grow back.

After the grass has been allowed to grow back to the original amount, Freddie then puts 3 cows into the paddock. This time he finds that it takes 7 days for the 3 cows to eat all the grass in the paddock. Freddie is puzzled that the cows took this long and consults a mathematician.

Freddie said "Geez mate, I dunno why me cows took that long to eat me paddock."
Marvin the mathematician replies "Well Freddie, tell me what assumptions you made."
Freddie replies "Well mate, maybe me cows got sick or somethin', cos I reckon that me 3 cows should have taken 6 days to eat me paddock, not 7 days! "
Marvin Replies, "I doubt that very much Freddie!"

After a while Marvin does some calculations and reveals that Freddie had overlooked an important assumption.  What was the assumption Freddie had overlooked?  Using Marvin's assumption, how long would a single cow take to eat the same paddock?

Source: Reader Ben Nie.  Bonus problem for homework.

Solutions were received from Alan O'Donnell, K Sengupta, Joseph DeVencentis, Mark Rickert, Nyles Heise, Philippe Fondanaiche, David Madfes, Dave Werder, Jeremy Galvagni, Earl Gose, Denis Borris, Dan Chirica, and Bernie R. Erickson.

Farmer Freddie forgot to account for the grass continuing to grow.  Many different solutions were received.  David Madfes' is very succinct:

The data presented is sufficient to calculate both the number of cow-day meals in the field before the cows are introduced and the rate at which the grass grows.
Let M = number of cow-day meals at time zero.
      g = amount of grass that grows in one day.
M + 3g = 18 meals   :  6 cows 3 days
M + 7g = 21 meals   :  3 cows 7 days
M = 63/4    The field starts with 15 3/4 cow-day meals.
g = 3/4       The grass grows at a rate of 3/4 of a cow-day meal per day.

[For one cow in X days, M + Xg = X meals]
If only one cow was let into the field it could eat for 63 days!

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