This puppy sparked a big debate for us. Take a swing at it

[Righty]

There were 1320 pumpkins in a pumpkin patch,but it was difficult for farmer Joe to find the perfect pumpkin.

Every third pumpkin was too small
Every fourth pumpkin was too green
Every fifth pumpkin had a broken stem
Every sixth pumpkin had the wrong shape

How many perfect pumpkins did farmer Joe find in the pumpkin patch?

[Wolfman]

Assuming no pumpking had 2 faults: 66

[Hoffs]

I would go for 440 perfect pumpkins.

A third, or 440, are too small.
A quarter, or 330, are too green. But of those, a third are also too small and have already been counted, leaving 220.
A fifth, or 264, are broken, but of those a half (440 + 220) are too small or too green, leaving 132.
A sixth, or 220, are wrong shape, but of those 60% (440+220+132) have another fault, leaving 88.

Therefore 440+220+132+88 have a fault, total 880. That leaves 440 good ones.

[Logan]

3 perfect pumpkins

[Righty]

See that is what I thought too, Hoffs. What if I told you the answer is 528 perfect pumpkins?

[Hoffs]

It does not surprise me at all to know that my answer is wrong; I was worried that I was not taking into account certain combinations. Unfortunately, I lack the ability to work it out even when told the answer.

[Alizaren]

3 perfect pumpkins

Hehe....I'm with Logan. I got about that far and figured all the others would cross each other off.  *giggles*

[Hettar]

See that is what I thought too, Hoffs. What if I told you the answer is 528 perfect pumpkins?

I got the same answer Hoffs did, I just got it a different way.  How is it 528 though   I can't work that answer into my equation no matter what I try...

[Alizaren]

My question is....is the answer really 528 or is Righty just baiting you all by saying "what if I told you the answer is 528 perfect pumpkins"?

*giggles*

[Kael]

My answer is same as hoffs

I never was a math genius mind you

[Righty]

No no baiting . I'll give you guys another day or two to work it out and then I'll share the answer we came up with.

[Alizaren]

Doh! I know how you got to the 528 ~

Can I say it???

[Alizaren]

Oh heck, I am going to say it. It's basically the way Hoffs figured it (or the way Hettar figured it - I am bogarting their idea,) however, you don't subtract ANY for the "every sixth had the wrong shape" because every sixth pumpkin would have been wiped out already because it was one of the "every third pumpkin was too small".

So, that would be 440+88=528

Is that it?

[blackwind Jeff]

Given that every 6th pumpkin is "an every 3rd pumpkin," you can eliminate any 6th pumpkin calculations (this is where Ali is on the money).  The idea of sieving out multiples of 3, 4, and 5 has already been executed by Hoffs, Righty, Ali, etc.  528 is most definitely the answer.

If you want to picture this without drawing a table of 1320 elements, you can reduce the sample size to 3*4*5, or 60, and then further to 30 by noticing repitition in this reduced set.  Conveniently, and necessarily, 30 is exactly 1/44 of 1320.  Sieving out multiples of 3, 4, and 5 out of 30 numbers is trivial and can be drawn up in no time.  This leaves 12 elements.  Our original set is 1320 pumpkins, and sieving the pumpkins out of 1320 is 12 * 44, or 528.

[Hoffs]

Ah well, I was misinterpreting the question. I was assuming that a random one third of the pumpkins were too small, a random quarter too green, etc. But if you are talking about just lining them up and counting off the faulty ones, then yes the answer is 528 and can be done very quickly.

Bah. Could have saved myself a lot of thinking if I had realised.