Duckweed doubles every day, and one plant covers the whole lake in 30 days. How long does it take if you throw in two plants? Most people say 15 days. The answer is 29. This lily pad problem is about exponential growth: when everything doubles daily, half the lake fills in the last day. Check it with a game, charts and five explanations.
The riddle is told with different heroes: duckweed, water lilies, algae. The point is the same. It shows how poorly we sense doubling, and why exponential growth looks harmless for so long.
One duckweed plant is thrown into a clean lake. Duckweed is a tiny plant that floats on the surface of the water.
Every day the amount of duckweed doubles: 1, 2, 4, 8, 16… After exactly 30 days it covers the whole lake.
Now two plants are thrown into an identical lake at once. The duckweed grows the same way.
How many days do two plants need to cover the whole lake? And on which day was the one-plant lake half covered?
"Twice the plants means twice as fast: 15 days"
In fact it takes 29 days. One plant becomes two after the first day. So two plants are only one day ahead.
Real duckweed doesn't grow this neatly: growth slows when space or nutrients run short. The model isolates the doubling itself.
Lock in your answer before reading on. One plant covers the lake in 30 days.
Your answer will be compared with the right one here.
Move the day slider or press Play. For three weeks the lake looks clear. Then in two or three days the duckweed takes almost all of it. The bright colour marks what grew in the last day: it is always half of all the duckweed.
| Day | Covered |
|---|
The chart shows the share of the lake covered when you start with 1, 2, 4 or 1,000 plants. All the curves have the same shape. Each one is shifted left by log₂ k days. Switch to a log scale and they become parallel straight lines.
Now each plant splits in two on a given day only with probability p. The computer grows 2,000 lakes from one plant and 2,000 lakes from two, and records the day each lake fills. At p = 100% you get the exact answer: 30 and 29 days.
Doubling multiplies the amount of duckweed. Here are five ways to see what follows from that.
The lake that got two plants looks on day 0 exactly like the first lake on day 1. From then on both lakes grow the same way, the second one a day ahead. So it fills on day 30 − 1 = 29.
| Day | Plants | Covered |
|---|
One plant, the same 30 days. On day 25 the duckweed takes 1/32 of the lake, about 3%. Half of the lake fills in the last day, a quarter in the day before.
Every doubling of the starting amount saves exactly one day. So 1,000 plants save only about 10 days, since 1,000 ≈ 2¹⁰.
If duckweed added the same area every day, two plants really would finish twice as fast. But it multiplies. In 15 days one plant becomes 2¹⁵ plants, and the full lake needs 15 more doublings. The only way to skip them is to start with 32,768 plants.
The brackets ⌈ ⌉ mean rounding up: we take the first whole day when the share reaches 100%. If k is a power of two, no rounding is needed. With 1,000 plants the lake is 97.7% covered on day 20, so it is full only on day 21.
Change the lake size or the growth rate and see how little the answer moves. Then check how much time extra room really buys.
The standard lake holds 2³⁰ plants: one doubling plant covers it in 30 days. A "×2" lake holds 2³¹. With tripling, each plant becomes three every day.
One plant, the standard lake. How much time is left until the lake is full from the moment the chosen share is covered?
A thought experiment by the physicist Albert Bartlett. Bacteria double every minute and fill a bottle at exactly noon. At 11:59 the bottle is half full. Imagine the bacteria find three more empty bottles. How much time does that buy?
| Time | Bacteria, in bottles | Space used |
|---|
The riddle appears in The Limits to Growth, the 1972 report to the Club of Rome by the team of Donella and Dennis Meadows. It is told there as a French children's riddle about a water lily that covers a pond in 30 days. The answer is the same: the pond is half covered on day 29.
In the legend, the inventor of chess asked the ruler for one grain on the first square, two on the second, four on the third and so on up to the 64th. That makes 2⁶⁴ − 1 ≈ 1.8 × 10¹⁹ grains. The last square alone holds more grains than all the previous squares together.
A 0.1 mm sheet folded 42 times would be 0.1 mm × 2⁴² ≈ 440,000 km thick. That's more than the average distance to the Moon, about 384,400 km. In reality paper can be folded in half only about a dozen times.
The physicist Albert Bartlett gave his lecture "Arithmetic, Population and Energy" more than a thousand times. The bacteria-in-a-bottle example comes from it. He held that the greatest shortcoming of the human race is our failure to understand the exponential function.
If something grows by r% a year, it doubles in about 70/r years. At 7% a year it doubles in about 10 years; at 2% a year, in 35. Bankers often use 72 instead of 70, because 72 divides neatly by 2, 3, 4, 6, 8 and 9.
Early in an epidemic, while almost everyone is still susceptible, case numbers often grow roughly exponentially. That is why a few days of delay at the start matter so much: every doubling time doubles the number of cases, like duckweed on the lake.
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