# Doubles and Trebles [closed]

There are two ponds having lotus flowers - pond A and pond B.

The property of the flowers of pond A is such that they get doubled (that is twice) in number every day as compared to previous day (6:00 AM - next day 6:00 AM is considered to be a day).

The property of the flowers of pond B is such that they get trebled(that is three times) in number every day as compared to previous day.

On the first day in a month having 30 days, it is observed that there is one lotus in each of the ponds and on 30th day, both the ponds are fully filled with flowers at the same time. Then the question is, which of the below events happened first

(1) Pond A getting 2/3 rd of its area filled with flowers

                          OR


(2) Pond B getting 1/2 of its area filled with the flowers

and what is the time difference between those two events (till minutes precision) ?

## closed as off-topic by Sconibulus, JonMark Perry, Gamow, Ankoganit, n_plumAug 22 '17 at 17:35

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – Sconibulus, JonMark Perry, Gamow, Ankoganit, n_plum
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• This seems like something you'd get in a maths exam paper... – Jamie Barker Aug 22 '17 at 13:56
• I think this is more calculation than puzzle. – Hugh Meyers Aug 22 '17 at 14:20
• Are they both filled at the same time? So on exactly Day 30 both ponds become full? Or is one overpopulated? – Dexter Aug 22 '17 at 14:31
• @Dexter - Yes, both of them filled at the same time. On day 30th both ponds become full. There were no cases of overfilling. Accordingly edited my question. – Mea Culpa Nay Aug 22 '17 at 14:40
• The solution bears out that this is rote application of math, not really a puzzle. (See, in particular, this answer which discusses why math problems, even "disguised as puzzles" as you put it, are still not actual puzzles —they're not fun to solve, and they lack that "a-ha!" moment puzzlers find so satisfying.) – Rubio Aug 23 '17 at 13:30

The areas of pond A and B are, respectively,

229 and 329 times the area of a lily pad, since there are 2n-1 lilies in pond A on day n and 3n-1 lilies in pond A.

If we approximate this as a continuous process, then the number of days at which pond A is two-thirds full will be given by the equation

$2^{n-1} = \frac{2}{3} 2^{29}$, which has a solution of $n = 29.4150375...$, which corresponds to 3:57.65 PM on day 29.

Doing the same for pond B, we have

$3^{n-1} = \frac{1}{2} 3^{29}$, which has a solution of $n = 29.3690702...$, which corresponds to 2:51.46 PM on day 29.

Thus,

Pond B becomes half-full approximately one hour and six minutes before Pond A becomes two-thirds full. This number, in days, is equal to $\log_2 \frac{2}{3} - \log_3 \frac{1}{2} = 0.04596725285...$

Also note that

Pond B is $3^{29}/2^{29} \approx 127,834$ times larger than Pond A. This means that if Pond A is the size of a tennis court, Pond B is about half the size of Manhattan.

• well done. Good one and fully illustrated. This was the expected solution. – Mea Culpa Nay Aug 23 '17 at 6:43
• @boboquack... those (fractions) are representing days and not for counts of lotus / lillies. – Mea Culpa Nay Aug 23 '17 at 11:41

Considering BOTH ponds are only completely full only on the 30th day (pond B being full much earlier than that) and both ponds have the same area. flowers in pond A = 2^(days-1) flowers in pond B = 3^(days-1) Meaning that the area of apond is 2^29 = 536,870,912 flowers Two thirds of that is 357,913,941 (rounded down from .33...) Half of that is 268,435,456 It would take pond A log2(357,913,941) = 28.4 days to reach 2/3rds of the area in flower cover. It would take pond B log3(268,435,456) = 17.66 days to reach half of the area in flower cover. Pond B will be covered half faster than pond A will be covered 2 thirds. But simple reasoning would already suggest that pond A can never overtake pond B anyway since pond B "grows" faster.

• Pond B has 1/3 of its area covered on the second last day. Reaching 1/2 should be in the last day, not the 17th. It is never stated that the ponds are the same size and is stated that they both are filled at the same time. – Apep Aug 22 '17 at 14:44
• @Gilles Lesire... it is not the question of overtaking ...but it is of differently filled sizes of ponds. – Mea Culpa Nay Aug 22 '17 at 14:50
• @Gilles Lesire, please check your answer. It seems to be a nope ! – Mea Culpa Nay Aug 22 '17 at 15:00