The lesson of the lily pond

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When will the lilies cover half the water surface?

Here is a simple math puzzle many people get wrong. Imagine you have built a swimming pond in the garden, and for decoration you drop a newly acquired lily in the middle. The vendor says you can expect that kind of lily to grow by doubling every day. So after one day there are two lilies, on day three there are four, then eight on the fourth day, and so on. On day thirty, you find that the entire pond is covered in lilies, and you can no longer swim. After how many days was the pond exactly half full of lilies (which is probably when the alarm bells would have rung in your mind that you need to pull some out so you can continue to swim in the pond)?

The most common answer is 15 days. (So, you could have cleared some of them out over a few days. No big deal.)

Unfortunately, that answer is wrong. The correct answer is 29 days! The day before the pond was completely full. (Too late!)

Videos of people explaining this problem and the solution have been all over social (and mainstream) media in recent months, to help people understand how viruses grow and spread, as authorities around the world struggled to help people understand the potential danger of the growth of the novel coronavirus. Because, in terms of the numbers, viruses grow and spread the same way as the lilies in the pond.

For a while it looks like the growth is slow, just 1 lily the first day, then 2 the next day, then just 4, and so on. If you were swimming around in the pond, for the first three weeks it would be easy to avoid bumping into a lily. Even after four weeks (28 days), 3/4 of the pond is still free of lilies. Plenty of time to act you might think. But the very next day, day 29, the lilies cover half the pond, and your time to act is up. One more day and the lilies have taken over the entire pond.

It’s what mathematicians call exponential growth. It’s the same kind of growth you find with climate change, which is why scientists — who do understand exponential growth — are so worried about it. With exponential growth, if preventative action is required, you have to do it early, when the problem seems trivially small, and there doesn’t seem to be anything to worry about. For instance, it would be easy to remove the lilies during the first three weeks when almost all the water surface is clear. If you leave it too long to act, it is too late.

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Exponential growth is sometimes said to exhibit a “hockey stick graph.” If you draw a graph of the growth, with time on the horizontal axis, the increase is at first imperceptible. Then suddenly the graph starts to kick up, getting steeper and steeper.

Understanding exponential growth is then, clearly an important life skill. The trouble is, our evolutionary history did not equip us to fully grasp it. We tend to think linearly — hence the common answer that the pond will be half full after 15 days. Yet there are ways to think about growth problems that enable us to make good decisions.

One way in particular can be very helpful: think backwards.

In the case of the lily pond, the crisis situation (when it is too late and you can no longer even get into the water, let alone swim) is day 30. So work backwards from that to see when you should act. Thinking backwards, when would the pond be half full? Well, the lilies double every day, so to go from full to half-full, you have to go back exactly one day.

Suddenly, a math problem that trips up many people becomes a trivial observation. Go back two days and the pond is only 1/4 full. Go back three days and it is only 1/8 full. And so on.

So what has this got to do with BrainQuake?

Look at any of the puzzles. In each case, there is a final state you want to achieve: all the items removed from the wheel in Gears, all the output tanks filled in Tanks, all the trays filled in Tiles. Next time you work on a puzzle, reflect on how you solve it. Maybe not for the first few puzzles, but after a while, you will find yourself picturing the final state (or the penultimate one) and asking yourself how you can get there. The app doesn’t tell you to do that; you just figure out that working backwards from the solution state is a smart way to start. Sure, you are going to have to work forwards to actually solve the puzzle. But your overall strategy will come from initially working backwards.

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A good way to start this Tiles puzzle is work backwards from the finish. Where would you need to place tiles so that with the growth patterns available, your final move would fill the trays? That might take you from the initial stage 1 to stage 4. Working backwards from 4 takes you to stage 3, and then to stage 2, which is where to place the tiles. Likely you don’t exactly do this. You probably alternate between backwards and forwards thinking. This kind of hybrid thinking is extremely common in advanced mathematical problem solving.

As I have said before on this blog, BrainQuake is designed to help people become better mathematical reasoners — better problem solvers. We don’t focus on the formulas, equations, and procedures of a typical school math class. Yes, you need all those to handle the details, and besides the school classroom there are plenty of math apps that will teach you how to do all of that. (In fact, your basic execution skills are likely to improve simply by using our app.) But for many problems in today’s world, getting the big picture and formulating an overall solution strategy are at least as important, and in some ways far more important. Especially since there are powerful, freely-available tools (like Wolfram Alpha) that can handle all the procedural stuff for you.

The best way to acquire that high-level math thinking ability — I suspect it is the only way — is by experience: getting practice dealing with problems where the key is to look at the problem the right way, and to combine basic math skills with high-level strategic thinking. In particular, becoming skilled at blending backward (= big picture) thinking and forward (= the steps of the calculation) thinking is the key to solving most of the major problems we face today (and many of the more minor ones for that matter).

Our puzzles are designed to provide a mental gymnasium to develop those critical mathematical thinking skills.

In fact, they provide an additional key learning experience. Many of our puzzles involve trying to find a solution that meets various constraints, sometimes contradictory ones (number of moves, number of items collected, order of items collection, number of tiles used, and the like). This too is a critical problem solving ability.

Yes, they are “just puzzles.” But so too is the lily pond puzzle, and mathematically that is exactly the pandemic growth problem governments around the world are grappling with right now.

Of course, your immediate goal, or that of your child, may not be to save the world, but to pass that next math exam, focused on basic procedural skills. But here’s the thing. As independent university studies have shown, solving BrainQuake puzzles results in improved results in traditional exams, particularly in the harder, multi-step questions that typically come towards the end of those tests. Knowing how to approach a problem — playing with it and looking at it in forward and backward directions — can make all the difference. Our recent societal experience with the current pandemic provides a dramatic, high stakes example. But passing an exam can be high stakes too!

– Keith

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Developing children’s true math proficiency

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