## Poll

17 votes (50%) | |||

13 votes (38.23%) | |||

5 votes (14.7%) | |||

2 votes (5.88%) | |||

11 votes (32.35%) | |||

3 votes (8.82%) | |||

6 votes (17.64%) | |||

5 votes (14.7%) | |||

10 votes (29.41%) | |||

8 votes (23.52%) |

**34 members have voted**

Quote:aceside

Apparently N=5, because when N=6, that pyramid will be flat.

Quote:Wizard

1. 5. At six, the pyramid would be flat. I'm not sure if that counts as a pyramid.

2. I'll bother to answer that if you confirm my answer of 5.

3. Based on a pentagonal base, 108.

I have a feeling I am not understanding the question correctly.

Quote:ChesterDog

Between the three of you, you are...

Correct!!!

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An old joke...

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.

Using Marvin's assumption, how long would a single cow take to eat the same paddock?

Quote:Gialmere

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.

Using Marvin's assumption, how long would a single cow take to eat the same paddock?

link to original post

The assumption is that the grass keeps growing.

Let x be the fraction of the grass each cow can eat in a day, and y be the fraction of the original grass that grows back in 24 hours.

18x - 2y = 1

21x - 6y = 1

18x - 2y = 21x - 6y

y = 3/4 x

18x - 2y = 1 -> x = 2/33 -> y = 1/22

One cow will eat 4/66 of the field each day, and 3/66 will grow back, so it takes 66 days.

In the first case 6 cows feed on Day 0, 1, 2 and 3. So there are, in total, 6x4=24 feeds.

In the second case 3 cows feed on Day 0, 1, 2, 3, 4, 5, 6 and 7. So there are, in total, 3x8=24 feeds.

The error of the farmer was assuming the cows ate at a constant rate, whereas they only eat once a day. In the first case there were 4 feeds for each cow (3 days), in the second, 8 feeds for each cow (7 days). For those not familiar with French, I think that's the logic they use for the word fortnight = "le quinze jours"!

Let:

i = initial amount of grass

g = grass grown in one day

e = grass eaten by one cow in one day

We are given:

i + 3g = 6 * 3 e

i + 7g = 3 * 7 e

In plain English, the first equation says that in 3 days 6 cows eat the initial amount of grass plus what was grown in 3 days.

The second equation says in 7 days 3 cows eat the initial amount of grass plus what was grown in 7 days.

So, we have two equations and three unknowns. That isn't good. However, we don't need to know i to get our answer. Let's assign i an arbitrary value of 100.

Now we have two equations and two unknowns. Some basic matrix algebra results in:

g = 100/21 =~ 4.7619

e = 400/63 =~ 6.3492

Let x = how long it takes one cow to clear the field.

i + xg = xe

x = i/(e-g) = 100/((400/63) - (100/21)) = 63

Putting in other values of i will still result in x = 63.

There is probably a more elegant way to do this, but this is how I did it.

Good problem!