#### Posers and Puzzles

davegage
Posers and Puzzles 09 May '05 04:37
1. 09 May '05 04:37
Suppose two farmers (farmer A and B) are planting two separate infinitely long rows of seeds. They work side by side, planting the seeds at the same rate.

At the same time, two birds (birds A and B) are trying to make their lives difficult. Bird A is following farmer A and Bird A is eating every tenth seed that farmer A throws down. Meanwhile, Bird B is following Farmer B, and after every tenth seed Farmer B throws down, Bird B eats the first remaining seed in Farmer B's row.

Thus the farmers are planting at the same rate, and the birds are eating at the same rate.

However, when farmer A "finishes" his row, there are infinitely many seeds left that Bird A did not eat; on the other hand, when farmer B "finishes" his row, there will be no seeds left in his row, for bird B will eat every one eventually.

How can this be?
2. AThousandYoung
All My Soldiers...
09 May '05 05:17
Originally posted by davegage
Suppose two farmers (farmer A and B) are planting two separate infinitely long rows of seeds. They work side by side, planting the seeds at the same rate.

At the same time, two birds (birds A and B) are trying to make their lives difficult. Bird A is following farmer A and Bird A is eating every tenth seed that farmer A throws down. Meanwhile, Bird B ...[text shortened]... will be no seeds left in his row, for bird B will eat every one eventually.

How can this be?
Bird B only eats a seed when Farmer B throws down ten seeds. If Farmer B stops throwing down seeds, Bird B will stop eating them...so Row B will only lose 1/10 seeds.
3. 09 May '05 05:451 edit
Originally posted by AThousandYoung
Bird B only eats a seed when Farmer B throws down ten seeds. If Farmer B stops throwing down seeds, Bird B will stop eating them...so Row B will only lose 1/10 seeds.
I suppose it's a little tricky to talk about a farmer "finishing" his planting.

Consider the following:

1) Suppose the farmers plant the first seed in 1 second, the second seed in 1/2 second, the third in 1/4 second, etc. Then they are done with their infinite rows in finite time (2 seconds). After these two seconds, are there any seeds left in farmer B's row?

2) Another way to approach it: the rows being infinite, the farmers never really finish their planting. In that case, Bird B eats (eventually) every seed he comes to, and thus leaves no seed uneaten. On the other hand, Bird A leaves 9 of every 10 seeds uneaten. How do you reconcile that?

EDIT: In keeping with the spirit of the original question, I am more interested in how you reconcile view 2) above...
4. 09 May '05 10:42
Originally posted by davegage
2) Another way to approach it: the rows being infinite, the farmers never really finish their planting. In that case, Bird B eats (eventually) every seed he comes to, and thus leaves no seed uneaten.
i dont get it. if farmer throws more seeds per time unit than bird eats, how can that bird ever eat all seeds? seems to me that bird will never catch farmer...
5. Palynka
Upward Spiral
09 May '05 12:121 edit
Originally posted by davegage
I suppose it's a little tricky to talk about a farmer "finishing" his planting.

Consider the following:

1) Suppose the farmers plant the first seed in 1 second, the second seed in 1/2 second, the third in 1/4 second, etc. Then th ...[text shortened]... estion, I am more interested in how you reconcile view 2) above...
If you fromulate this question in series, you'll see that the number of seeds remains infinite.

Edit: Mathematical series
6. 10 May '05 02:22
This kinda reminds me of "what is infinity minus infinity?"

is it zero by normal math? or negative infinity because if you take an unlimited amount away, you'd get down to an unlimited amount.

smart calculators will say it's undefined though.
7. 16 May '05 02:39
Originally posted by Palynka
If you fromulate this question in series, you'll see that the number of seeds remains infinite.

Edit: Mathematical series
This is a classic probability problem. People don't understand it, because they tend to think of the process as finite, but it is not. Bird A will be leaving behind 9 seeds for every unit of planting time that goes on. So, Bird A eats seed 1, and then seeds 2 through 10 are safe forever, because it will then go after seed 11, then 21, and so on. Bird B is not leaving behind any seeds. So, even though the farmer is planting seeds 10 times as fast, as the bird is eating the seeds, the n_th seed will be eaten after 10*n units of planting time have passed. Therefore, every seed in the B plot is going to get eaten at some point.
8. 16 May '05 05:05
First of all I'd like to say that infinity is not a number, so you cannot make mathematical calculations with it. It's no more valid to say what is f(infinity) than it is to say what is f(Michael). It just doesn't make sense.
Second of all, since you say there are infinity seeds that will be planted, that automatically messes up the problem not only because of the above stated reason but because the bird will never catch up with eating the seeds if the farmer never stops.
Third of all, adding in a factor of time by saying it takes 2 seconds to complete the process doesn't change a thing. All it does is direct one towards thinking finitely because the event seems to have an end in time [But as far as the process goes those 2 seconds are eternity. You can probably find theory on this if you search the web but the reason this is so is you must divide (remember this is invalid, because you cannnot use any functions of infinity) the period of time into infintesimally small units to express the time it took to plant one seed after another] when it actually does not [in fact, associating the word infinity with something like time or space is invalid, because it cannot exist in time or space. The only thing infinite is God, who is not limited by spacetime, that is why God is eternal (it is more accurate to say God is beside time began than before time began), in fact he created time itself (and everything else for that matter). So numbers (limited) are entirely different from infinity (unlimited). To fully grasp infinity and eternity, wait until you die and reach eternal life/death. You'll definitely understand it all then and so will I (I'm just stating the theory)
Fourth of all, since the farmers never stop planting, saying after they finish simply doesn't make sense. When you add the quotes: '"finish"' you must mean at any given point in the process of planting (in which case there would be the same amount of seeds on the ground for each farmer). So far all this would be resolved by saying the farmers plants, for example, a 100 long row of seeds. It is illogical to say it is never ending [they must be doing it in eternity (heaven/hell) to have a limitless amount of what we would understand as space and time to do this]
Fifth of all, there is only one tenth seed ðŸ˜‰ just like there's only one first seed (well, ok I guess it could literally mean what you wanted it to mean as well)
Sixth of all the word eventually doesn't make sense in the context (not too dificult to figure out)
Seventh of all if for example they were doing 100 long rows of seed, each would have the same amount of seeds at the end if they simply replaced the seeds the birds ate ðŸ˜‰
And finally, if for example they were 100 long rows of seed there should be 90 seeds left for farmer A and 9 seeds left for farmer B (unless the birds pooped the seeds back on the groundðŸ˜›)
9. Palynka
Upward Spiral
16 May '05 09:55
Originally posted by rheymans
This is a classic probability problem. People don't understand it, because they tend to think of the process as finite, but it is not. Bird A will be leaving behind 9 seeds for every unit of planting time that goes on. So, Bird A eats seed 1, and then seeds 2 through 10 are safe forever, because it will then go after seed 11, then 21, and so on. Bird B ...[text shortened]... ting time have passed. Therefore, every seed in the B plot is going to get eaten at some point.
I agree, but:

Quoting from the original post
there will be no seeds left in his row, for bird B will eat every one eventually.

The second conclusion is correct, but the first one is not.
10. PBE6
Bananarama
16 May '05 14:00
Originally posted by bobbob1056th
The only thing infinite is God, who is not limited by spacetime, that is why God is eternal (it is more accurate to say God is beside time began than before time began), in fact he created time itself (and everything else for that matter). So numbers (limited) are entirely different from infinity (unlimited). To fully grasp infinity and eternity, wait until you die and reach eternal life/death.
I wish you had understood infinity and eternity before you posted that Jackson Pollock of an answer.

And shouldn't that religious diatribe be sent to the Religious Debates forum? I'm sure you and Ivanhoe would have fun debating the issue until an infinite number of cows came home.
11. 16 May '05 15:19
Originally posted by bobbob1056th
First of all I'd like to say that infinity is not a number, so you cannot make mathematical calculations with it. It's no more valid to say what is f(infinity) than it is to say what is f(Michael). It just doesn't make sense.
Second of all, since you say there are infinity seeds that will be planted, that automatically messes up the problem not ...[text shortened]... farmer A and 9 seeds left for farmer B (unless the birds pooped the seeds back on the groundðŸ˜›)
I agree.....with almost nothing you have stated.

What a mess of a post.

I wouldn't know where to begin.
12. 16 May '05 15:191 edit
Originally posted by Palynka
The second conclusion is correct, but the first one is not.
Prove this statement, if you please...specifically for the case where they finish planting in finite time.
13. 16 May '05 15:31
Originally posted by bobbob1056th
First of all I'd like to say that infinity is not a number, so you cannot make mathematical calculations with it. It's no more valid to say what is f(infinity) than it is to say what is f(Michael). It just doesn't make sense.
Second of all, since you say there are infinity seeds that will be planted, that automatically messes up the problem not ...[text shortened]... farmer A and 9 seeds left for farmer B (unless the birds pooped the seeds back on the groundðŸ˜›)
Ugh.

The post is even worse on the second reading.
14. Palynka
Upward Spiral
16 May '05 15:41
Originally posted by davegage
Prove this statement, if you please...specifically for the case where they finish planting in finite time.
Use a subtraction of the two diverging series and you'll see that the resulting series would diverge.

Therefore all seeds would be eaten (the subtracting series is divergent) but there would always be seeds (the resulting series is divergent).
15. 16 May '05 16:20
Originally posted by Palynka
Use a subtraction of the two diverging series and you'll see that the resulting series would diverge.

Therefore all seeds would be eaten (the subtracting series is divergent) but there would always be seeds (the resulting series is divergent).
I agree that all the series you are refering to diverge -- that is clear since the assumption we are working with is that the farmers' rows are infinite. But I don't think this answers the question, which deals primarily with cardinality.

Consider again the case where the farmers plant the first seed in 1 second, the second seed in 1/2 of a second, the third seed in 1/4 of a second. Then the time it takes them to plant their rows converges to 2 seconds.

Then what is wrong with the following deduction:

Premise 1. Bird B will eventually eat all of the seeds in his row.
Premise 2. 'Eventually' is reached after 2 seconds.
Premise 3. Therefore, after 2 seconds, there are no seeds remaining in row B.

You already said you agree with Premise 1. So if you reject Premise 3, then what is wrong with Premise 2?