The Doubling Puzzle: On the left two unhappy sheep stand before one patch of grass. Ten days later, the two sheep seem happy and are standing before two patches of grass.

The doubling puzzle has been popular among math enthusiasts for centuries now. In this article, I will be presenting a simplified version of this puzzle. It is by no means a very hard puzzle, but it does require critical and coordinated thinking to solve.

Once we have solved the puzzle, I will be covering the dynamics behind such puzzles and explain how solving such puzzles could come in handy in day-to-day life.

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The Clever Farmer

Our puzzle starts with a clever farmer who has always been fond of numbers. You see, unlike the rest of his farmer friends, he wants to take advantage of his affinity to numbers.

Currently, he is in need of raising grass in one of his plots. The faster he gets this job done, the more beneficial it would be for his farm animals. He currently buys the necessary grass at four times the price of what it would cost him to raise the grass himself. Since he has an empty plot, he goes in search of suitable grass patches that he can just plant.

The Doubling Puzzle

In the local farmer’s market, he comes across a dealer selling a special type of grass. This dealer tells him that a single patch of this special grass doubles in area every day! The farmer has known this dealer for decades and trusts him completely. He quickly whips out his notebook and works out that if he plants one patch of this special grass, it would completely cover his plot in 10 days.

The Doubling Puzzle: On the left two unhappy sheep stand before one patch of grass. Ten days later, the two sheep seem happy and are standing before two patches of grass.
Image created by the author

He is considering getting 4 such grass patches to really accelerate the process. The only challenge is his budget. The special grass is very expensive. He has to buy grass externally for his cattle until his plot is fully covered with the fresh plantation. Therefore, he wants to convince himself that buying 4 grass patches is worth the speed before he commits.

He stands there staring at his notebook wondering if he should go with 1 patch or 4 patches of the special grass. This is the puzzle!

Take a moment (but not too long) and ponder upon what your mathematical intuition says. Hold that thought. If you are the clever type who can do quick calculations in the head, feel free to do it and hold that answer in your head.

Solving the Doubling Puzzle

There are two ways in which we can approach this problem: the slower approach and the faster approach. The faster approach requires quick abstraction, and some people may not appreciate or wish to do this. The slower approach on the other hand is more intuitive and satisfying. It also provides more information in the answer than the faster approach. So, I’ll go ahead and present both approaches. The key to both approaches, however, is thinking quickly in multiples of 2, as we are dealing with doubling here.

The Slower Approach

The first major clue in the puzzle comes from the farmer’s quick calculation. He has already computed that if he buys 1 special grass patch, it would take 10 days for his plot to be completely covered. From this, we could work out how many patches his plot is comprised of. We could start with 2 and double it ten times to get the result.

Solving the doubling puzzle: a table with days from 1 to 10 listed in the top row and the number of grass patches listed in the bottom row. Day 1 has 2 patches, day 2 has 4 patches, and so on upto day 10, which has 1024 patches (the number of patches doubles everyday)
Math illustrated by the author

So, it turns out that the farmer’s plot has 1024 grass patches to be filled in 10 days. A faster way to work out this answer would be to just compute the value of 2¹⁰. In doubling problems, the frequency of doubling can be the exponent with 2 as the base. And yes, it helps if one can memoriz powers of 2 up to at least 10 for quick computations.

Now the next question is: Is it worth buying 4 such special grass patches? Now recall your mathematical intuition or your computed answer from earlier. Let’s apply the same approach as before and see how fast we fill 1024 grass patches when we buy 4 patches instead of 1.

Solving the doubling puzzle: A third row is added to the table from before. This row starts with 4 patches, and therefore, there are 8 patches on day on. From thereon, the number of patches double per day as before, and 1024 patches are reached on day 8.
Math illustrated by the author

It turns out that the farmer saves just 2 days by quadrupling his expenses by buying 4 grass patches instead of 1. Given the farmer’s circumstances, the extra expense is better avoided! Did this result surprise you? Or did your mathematical intuition serve you well? Before you veer off to the comments section, let us look at the faster approach.

The Faster Approach

The faster approach is not concerned with the minute details. It is aimed at computing one thing only: the number of days required by 4 patches to populate the plot. Any other calculation is considered a waste of time in this approach.

Solving the doubling puzzle faster: On the left, there is one patch of grass. After 10 days, on the right, there are 2 patches of grass.
Image created by the author

Furthermore, in this approach, we start from the end instead of the beginning. We know that 1 patch of special grass covers the plot in 10 days. That means that at end of 9 days, only half of the plot was covered. Similarly, at the end of 8 days, only a quarter of the plot was covered.

If it takes 1 patch of special grass 8 days to cover one-quarter of the plot, it takes 4 such patches the same number of days to cover the entire plot (4-quarters in total).

Therefore, we arrive at the same answer faster. We didn’t bother with the minute details with this approach, but that is a price we knew we were paying.

The Doubling Dynamics

The doubling dynamics involved in such puzzles is innately counter-intuitive to typical human thinking. Therefore, we are not able to process the information straight away. In fact, our natural approach is to extrapolate linearly, whereas the multiplicative dynamics are non-linear.

For those of you whose mathematical intuition was off the mark, this is likely to be one of the main reasons. If you are one of those people who were able to think non-linearly straight away, be aware that this may come at the cost of over-complicating daily situations. Human beings think linearly for a reason — it is arguably the most efficient way of going about everyday life (evolutionarily speaking).

Having said this, there exists a potential sweet spot — using heuristics. As soon as you are able to identify doubling dynamics, you should set 2 as the base and the frequency (number of days in this case) as the exponent. Similarly, other multiplicative dynamics would use corresponding bases (tripling — 3; quadrupling — 4, etc.).

Real-Life Applications of the Doubling Puzzle

Any real-life phenomenon that involves growth or decay features multiplicative dynamics. Consequently, if you are able to recognize and apply the above heuristic when encountering such phenomena, you are likely to gain a positive outcome. Typical examples of such phenomena include financial investing, consumption of resources from a repository, projection of purchase based on consumption, travel planning, etc.

We have now arrived at the end of this article. You started by tackling a tricky little math puzzle. You then got to know the dynamics behind such puzzles. Finally, you got to see how you can apply this knowledge to your benefit in real-life applications. I wish you happy mathematical adventures in your journey ahead!

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Further reading that might interest you: How To Use Science To Answer: “Should You Walk Or Run In The Rain?” and Why Do You See Mirrors Flipping Words?

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