- 20 Jul '06 13:50There 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? - 20 Jul '06 15:15

Still 40 days?*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? - 20 Jul '06 15:54

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.*Originally posted by crazyblue***Still 40 days?** - 20 Jul '06 19:17 / 5 edits

DayLeavesLeaves*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?

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**

40**549,755,813,888**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! - 21 Jul '06 00:32Wow... 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! - 21 Jul '06 08:38Okey, 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. - 21 Jul '06 08:48Wow... 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'?