The Leaves & The Pond.

ckoh1965
Posers and Puzzles 20 Jul '06 13:50
1. 20 Jul '06 13:50
There is this type of leaf that grows on the surface of a pond. It happens that the leaf multiplies every 24 hours. Therefore 1 becomes 2 the next day; then becomes 4 after the second day; becoming 8 the day after and so on and so forth.

Now it has been determined that it takes 40 days to cover up the whole surface of the pond if we start day 1 with one leaf only. OK, fine, now we take away all those leaves and start over again, this time with 2 leaves on day 1. How long does it take to cover the entire surface of the pond?
2. 20 Jul '06 15:15
Originally posted by ckoh1965
There is this type of leaf that grows on the surface of a pond. It happens that the leaf multiplies every 24 hours. Therefore 1 becomes 2 the next day; then becomes 4 after the second day; becoming 8 the day after and so on and so forth.

Now it has been determined that it takes 40 days to cover up the whole surface of the pond if we start day 1 with one ...[text shortened]... this time with 2 leaves on day 1. How long does it take to cover the entire surface of the pond?
Still 40 days?
3. 20 Jul '06 15:17
No.
4. 20 Jul '06 15:32
39 methinks.
5. 20 Jul '06 15:50
20 days
6. 20 Jul '06 15:54
Originally posted by crazyblue
Still 40 days?
I would of thought 40 as well. That is if we are dealing in discrete numbers of days. Because starting with two leaves we get to the same amount as 40 days with 1 leaf starting, but minus the one that started out. So it would take that extra day just to fill the pond. But as 40 is wrong I'm going for the obvious 39.
7. 20 Jul '06 15:54
How do you keep an idiot in suspense?
8. 20 Jul '06 19:175 edits
Originally posted by ckoh1965
There is this type of leaf that grows on the surface of a pond. It happens that the leaf multiplies every 24 hours. Therefore 1 becomes 2 the next day; then becomes 4 after the second day; becoming 8 the day after and so on and so forth.

Now it has been determined that it takes 40 days to cover up the whole surface of the pond if we start day 1 with one ...[text shortened]... this time with 2 leaves on day 1. How long does it take to cover the entire surface of the pond?
DayLeavesLeaves
112
2 2 4
3 4 8
4 8 16
5 16 32
6 32 64
7 64 128
8 128 256
9 256 512
10 512 1,024
11 1,024 2,048
12 2,048 4,096
13 4,096 8,192
14 8,192 16,384
15 16,384 32,768
16 32,768 65,536
17 65,536 131,072
18 131,072 262,144
19 262,144 524,288
20 524,288 1,048,576
21 1,048,576 2,097,152
22 2,097,152 4,194,304
23 4,194,304 8,388,608
24 8,388,608 16,777,216
25 16,777,216 33,554,432
26 33,554,432 67,108,864
27 67,108,864 134,217,728
28 134,217,728 268,435,456
29 268,435,456 536,870,912
30 536,870,912 1,073,741,824
31 1,073,741,824 2,147,483,648
32 2,147,483,648 4,294,967,296
33 4,294,967,296 8,589,934,592
34 8,589,934,592 17,179,869,184
35 17,179,869,184 34,359,738,368
36 34,359,738,368 68,719,476,736
37 68,719,476,736 137,438,953,472
38 137,438,953,472 274,877,906,944
39 274,877,906,944 549,755,813,888
40549,755,813,888 1,099,511,627,776

I'll go out on a limb and say, oh, 39 days.
EDIT: Tried to get the columns to match. No luck!
9. 21 Jul '06 00:07
All this trouble over the least significant bit of mathematics.
10. 21 Jul '06 00:32
Wow... you clever people! Yes, the correct answer is 39 days, although I have a much simpler explanation that the one offerred by Freaky.

The most significant point is that the leaves double in number each day. This means that when we started with one leaf, because it took 40 days to cover the whole surface, then it must mean that on the day prior to that, it was only half covered. It follows that on the 40th day, that half doubled up to cover the entire surface.

OK, fine, now we imagine starting with 2 leaves. One of them will cover half of the surface in 39 days; and the other leaf does the same for the other half of the pond. Therefore, those 2 leaves cover the entire pond surface in 39 days!
11. 21 Jul '06 08:38
Okey, after 40 days we have 549,755,813,888 leaves covering the surface of the pond. Thats half a trillion leaves. !!!

Consider the every leaf weigh about 10 grammes. This make 5 trillion gramms or 50 billion kilograms or 50 million tonnes of leaves. !!!

Say we have a pond of 10 by 10 metres, 100 square metres and every leave have a surface of say 10 square centimetres. then you need 1000 leaves on every square metre. With half a trillion leaves you have to stack them half a billion leaves on top of eachother. !!!

Say every leaf has a thickness of 1 millimetre, then the height of this stack of leaves reaches half a million metres, or 500 kilometres in height. That's outside the atmosphere up to low orbit satellites. !!!

I don't think this scenario is plausible.
12. 21 Jul '06 08:48
Wow... your thoughts are really very, very far! I must admit that I didn't go that far myself. But the point of the problem is on the rational means on how to determine the number of days to cover the pond, given an earlier 40 days for a particular circumstance. I could have easily said it took, say, 10 days only to cover up the entire surface of the pond, and perhaps then it would be more realistic? In that sense, I guess my original question was a 'mistake'?
13. 21 Jul '06 09:02
Well, it is a good puzzle, and good puzzles don't have to be plausible.

More of the kind!
14. Bowmann
Non-Subscriber
21 Jul '06 21:431 edit
What a bunch of twonks!

Applying minimal lateral thinking, this puzzle is solved in one second.
15. 22 Jul '06 14:09
Originally posted by Bowmann
What a bunch of twonks!

Applying minimal lateral thinking, this puzzle is solved in one second.
It took my computer less than one second.