I posted this imaginary scenario and poll on Twitter.

The scenario was inspired by a classic problem used in psychological research on insight problem solving. Here is the classic version of the problem:^{1}

Water lilies double in area every 24 h. At the beginning of summer there is one water lily on the lake. It takes 60 days for the lake to become completely covered with water lilies. On which day is the lake half covered? (Sternberg and Davidson, 1982).

I only realized later that I changed the wording to ‘pond’ instead of ‘lake’. I had misremembered the wording. I should acknowledge this because it may have affected the outcome of the poll. However, it does not matter for my point, which is to ask why the psychologists thought the original scenario was a realistic one.^{2}

How big does a lake need to be for it to take 60 days for it to be covered by ‘doubling’ water lilies? The right answer is: about ten times the surface of the earth. To see this, let’s calculate. First we start with a reasonably sized water lily, say, on the order of 10 ☓ 10 cm^{2}. If it doubles for 60 days, the surface that the lilies will cover is 2^{59 }☓ 100 cm^{2} = 5.76 ☓ 10^{9 }km^{2}. The total surface area of Earth is about 510 million km^{2 }= 5.10 ☓ 10^{8} km^{2}.

When I posted the twitter poll, some people asked: But how large should I imagine the water lily?

It does not really matter, because for any reasonably sized water lily the surface of the pond would be unrealistically large; hence, my answer on Twitter above.

Say the water lily would be half the size of what you imagined it. Then after one day of doubling, the covered area would be again the same. To really make a difference in the scenario the water lily must be tiny. How tiny? As it turns out, it needs to be extremely tiny. Even if the water lily would be as small as 0.001 ☓ its original size, still the size of the ‘pond’ would need to be quite large for it to take 60 days to be covered. Namely, 5.67 ☓ 10^{6 }km^{2}, which is a fifth of the largest sea on earth (Philippine Sea, 24.8 ☓ 10^{6 }km^{2}), and more than half the surface of Europe (10.1 ☓ 10^{6 }km^{2}) or the surface of the USA (9.8 ☓ 10^{6 }km^{2}).

But just like people may underestimate the surface covered by doubling lilies, they may overestimate the impact of the water lily’s size on the area that will be covered after 60 days. To investigate this I posted a follow-up twitter poll.

To make it more straightforward to compare the results of the two polls, I have plotted the results side by side below.Of course, these results are from an uncontrolled twitter poll and thus should be taken with a grain of salt. Be that as it may, I think some interesting observations can be made. First, almost half of the people answering the poll seemed to be in the right ball park with their intuition about the size of the pond. This was an unexpected outcome for me, because years ago I ran this scenario as part of a controlled experiment (never published it, unfortunately) with a different population (i.e., university students) and then the median response was ‘soccer field’. Possibly the poll results show a different pattern because a disproportionate number of my twitter followers having had some formal training in exponential growth.^{3} Second, about 30% of the people intuited the size of the pond to be anywhere between a bath tub and a lake for the normal lily, and about 42% for the tiny lily. This is a non-neglible number of people who substantially underestimated the size of the pond in the scenarios.

Why did I post these polls? I am a computational cognitive scientist and I am interested in the computational scope and limits of cognition. Exponential growth plays a central role in this research. I conjecture that some of the intuitions that may be thwarted by the water lily scenarios also come into play when cognitive scientists debate about what resource-bounded minds/brains can or cannot realistically compute. I plan to say more about this in a later blog. For now, I hope you enjoyed the water lily puzzles 🌺 😀

^{1 }Sternberg, R. J., & Davidson, J. E. (1982). The mind of the puzzler. *Psychology Today, 16*, 37-44. I noted on the wikipedia page for ‘exponential growth’ that Sternberg and Davidson’s scenario may in turn have been inspired by Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972). *The Limits to Growth*. New York: University Books.

^{2 } Thanks to Karl Teigen from University of Oslo, Norway, for pointing this out to me years ago. I do not know to what extent Sternberg and Davidson (1982) thought the scenario was realistic, or perhaps realized that it wasn’t but went with it regardless.

^{3 }Consistent with this interpretation, the bimodal nature of the distribution (most apparent for the tiny lily, but this may be because that manipulation served to drive the two modes apart) seems to show that there are two different populations sampled in this poll: people who understand exponential growth and people who do not. Possibly, the people with formal training in exponential growth quickly conjured up the multiplier 2^{60 }in their minds (not realizing that day 1 had only 1 lily) and then let this number govern their intuitions (I made this mistake at first as well!). If indeed day 1 is counted as day 0 then the water lilies in the first scenario would cover 2^{60 }☓ 100 cm^{2} = 1.2 ☓ 10^{10 }km^{2} which would be 100 times the surface of the earth. In the second scenario, the tiny water lilies would then cover 1.2 ☓ 10^{7 }km^{2}, which would be more than the surface of Europe (10.1 ☓ 10^{6 }km^{2}) or the surface of the USA (9.8 ☓ 10^{6 }km^{2}), and close to the surface of the Arctic Ocean (1.4 ☓ 10^{7} km^{2}). This would explain why estimates for this group are actually a bit on the high side.