1/
Get a cup of coffee.
In this thread, I'll show you how to count the number of oranges in a pile.
Believe it or not, this exercise can teach us a few lessons about investing.
Get a cup of coffee.
In this thread, I'll show you how to count the number of oranges in a pile.
Believe it or not, this exercise can teach us a few lessons about investing.
2/
Earlier this week, I conducted a poll, where I asked the following question:
Here's a pile of oranges, of height 4. The pile contains 20 oranges.
Along the same lines, suppose we build a bigger pile -- of height 100. How many oranges will this bigger pile contain?
Earlier this week, I conducted a poll, where I asked the following question:
Here's a pile of oranges, of height 4. The pile contains 20 oranges.
Along the same lines, suppose we build a bigger pile -- of height 100. How many oranges will this bigger pile contain?
3/
The right answer is 171,700. That's how many oranges there'll be in a (similarly constructed) pile of height 100.
The right answer did receive the most votes. Yay!
But a *majority* of respondents (~66.3%) got the answer wrong.
Note: *Plurality* does not mean *majority*!
The right answer is 171,700. That's how many oranges there'll be in a (similarly constructed) pile of height 100.
The right answer did receive the most votes. Yay!
But a *majority* of respondents (~66.3%) got the answer wrong.
Note: *Plurality* does not mean *majority*!
4/
Since most poll respondents got the answer wrong, I thought it might be useful to go through a few ways to approach such "counting exercises".
Hence this thread.
Since most poll respondents got the answer wrong, I thought it might be useful to go through a few ways to approach such "counting exercises".
Hence this thread.
5/
Generally, when we want to count a big collection (like a large pile of oranges), it's a good idea to build intuition by deconstructing a smaller "test case".
For example, we could start with a pile of height 4. And try to spot a pattern that extends to a pile of height 100.
Generally, when we want to count a big collection (like a large pile of oranges), it's a good idea to build intuition by deconstructing a smaller "test case".
For example, we could start with a pile of height 4. And try to spot a pattern that extends to a pile of height 100.
6/
In this spirit, I went to Costco and bought a box of oranges.
You know, it's surprisingly hard to pile them up. The moment we tack on oranges at the top, the ones at the bottom roll away and the pile collapses!
In this spirit, I went to Costco and bought a box of oranges.
You know, it's surprisingly hard to pile them up. The moment we tack on oranges at the top, the ones at the bottom roll away and the pile collapses!
7/
Anyway, after a few tries, I finally managed to build a somewhat stable pile of height 4.
From the pile's construction, the logic becomes immediately clear.
The bottom-most layer has 1+2+3+4 = 10 oranges.
The layer immediately above it has 1+2+3 = 6 oranges.
And so on.
Anyway, after a few tries, I finally managed to build a somewhat stable pile of height 4.
From the pile's construction, the logic becomes immediately clear.
The bottom-most layer has 1+2+3+4 = 10 oranges.
The layer immediately above it has 1+2+3 = 6 oranges.
And so on.
8/
We can now extrapolate this logic to a pile of height H, where H is any height we like (50, 100, 1000, etc.):
We can now extrapolate this logic to a pile of height H, where H is any height we like (50, 100, 1000, etc.):
9/
So, to get the total number of oranges in the pile, we should just add up the numbers in the last column above.
But before getting to that, we can apply a few simple ideas to get *rough estimates* of how many oranges our pile of height 100 will contain.
So, to get the total number of oranges in the pile, we should just add up the numbers in the last column above.
But before getting to that, we can apply a few simple ideas to get *rough estimates* of how many oranges our pile of height 100 will contain.
10/
For example, using the logic above, Layer 1 (the bottom-most layer) in our pile of height 100 will contain 1+2+ ... +100 oranges.
The average of these 100 numbers (1 to 100) is roughly 50. So, 1+2+ ... + 100 should roughly equal 50*100 = 5,000.
For example, using the logic above, Layer 1 (the bottom-most layer) in our pile of height 100 will contain 1+2+ ... +100 oranges.
The average of these 100 numbers (1 to 100) is roughly 50. So, 1+2+ ... + 100 should roughly equal 50*100 = 5,000.
11/
So, there will be ~5,000 oranges in *just* the bottom layer of our pile.
And yet, ~13.1% of poll respondents said there will be less than 1,000 oranges in our *whole* pile!
So, there will be ~5,000 oranges in *just* the bottom layer of our pile.
And yet, ~13.1% of poll respondents said there will be less than 1,000 oranges in our *whole* pile!
12/
Also, we know that the bottom layer has the *most* oranges. As we go up, the number of oranges per layer only goes down.
So, our pile cannot possibly contain more than about 5,000*100 = 500,000 oranges.
Yet, ~28% of respondents said it will contain more than a million!
Also, we know that the bottom layer has the *most* oranges. As we go up, the number of oranges per layer only goes down.
So, our pile cannot possibly contain more than about 5,000*100 = 500,000 oranges.
Yet, ~28% of respondents said it will contain more than a million!
13/
So far, we've determined that our pile has more than 5,000 but less than 500,000 oranges.
This is great. We've eliminated 2 of the 4 poll choices.
But can we do better? After all, 5K to 500K is kind of a wide range.
So far, we've determined that our pile has more than 5,000 but less than 500,000 oranges.
This is great. We've eliminated 2 of the 4 poll choices.
But can we do better? After all, 5K to 500K is kind of a wide range.
14/
It turns out we can.
Imagine someone has placed a bunch of oranges on the ground exactly 1 meter apart from one another.
Suppose we wanted to find out how many oranges are contained inside the blue circle below, whose radius is ~10 meters.
It turns out we can.
Imagine someone has placed a bunch of oranges on the ground exactly 1 meter apart from one another.
Suppose we wanted to find out how many oranges are contained inside the blue circle below, whose radius is ~10 meters.
15/
Intuition suggests that the number of oranges inside the blue circle should roughly equal the area of the circle (in square meters).
That would be "pi r squared", or roughly 3.14 * 10 * 10 = 314 oranges inside the circle.
Intuition suggests that the number of oranges inside the blue circle should roughly equal the area of the circle (in square meters).
That would be "pi r squared", or roughly 3.14 * 10 * 10 = 314 oranges inside the circle.
16/
The same idea can be used to roughly estimate the number of oranges in our pile of height H as well.
Our pile has the shape of a "regular tetrahedron" of height H.
The volume of this tetrahedron should roughly equal the number of oranges in our pile.
The same idea can be used to roughly estimate the number of oranges in our pile of height H as well.
Our pile has the shape of a "regular tetrahedron" of height H.
The volume of this tetrahedron should roughly equal the number of oranges in our pile.
17/
As we'll see, the estimate above is not super accurate.
But there's one key insight to be gleaned from this tetrahedron idea: the number of oranges in our pile *scales* cubically with height.
That is, the number of oranges in a pile of height H is proportional to H^3.
As we'll see, the estimate above is not super accurate.
But there's one key insight to be gleaned from this tetrahedron idea: the number of oranges in our pile *scales* cubically with height.
That is, the number of oranges in a pile of height H is proportional to H^3.
18/
So far, we've obtained *bounds* (eg, between 5K and 500K) and *rough estimates* (eg, ~117,851) for the number of oranges in our pile.
But is it possible to get an *exact* answer?
Indeed, it is.
So far, we've obtained *bounds* (eg, between 5K and 500K) and *rough estimates* (eg, ~117,851) for the number of oranges in our pile.
But is it possible to get an *exact* answer?
Indeed, it is.
19/
To arrive at the exact answer, we'll need a couple of formulas.
Formula 1: The sum of the first "n" natural numbers, ie, 1 + 2 + 3 + ... + n, and
Formula 2: The sum of the *squares* of the first "n" natural numbers, ie, 1^2 + 2^2 + 3^2 + ... + n^2.
To arrive at the exact answer, we'll need a couple of formulas.
Formula 1: The sum of the first "n" natural numbers, ie, 1 + 2 + 3 + ... + n, and
Formula 2: The sum of the *squares* of the first "n" natural numbers, ie, 1^2 + 2^2 + 3^2 + ... + n^2.
20/
Armed with these formulas, we can now calculate the exact number of oranges in our pile of height 100. It's 171,700.
Calculations:
Armed with these formulas, we can now calculate the exact number of oranges in our pile of height 100. It's 171,700.
Calculations:
21/
Not only that, we can *prove* that the number of oranges in a pile of height H will be exactly equal to (H*(H+1)*(H+2))/6 -- no matter what H is (4, 100, 1000, etc.).
Proof:
Not only that, we can *prove* that the number of oranges in a pile of height H will be exactly equal to (H*(H+1)*(H+2))/6 -- no matter what H is (4, 100, 1000, etc.).
Proof:
22/
Counting oranges may seem like an exercise without much practical utility.
But the process we went through above can teach us a number of lessons.
Lessons that are relevant to investing, and to life in general.
Counting oranges may seem like an exercise without much practical utility.
But the process we went through above can teach us a number of lessons.
Lessons that are relevant to investing, and to life in general.
23/
Key Lesson 1. Life sometimes offers up no-brainers. We should be ready to capitalize.
For example, with only a cursory analysis, we could eliminate 2 out of the 4 poll choices (less than 1000, and more than 1,000,000). But ~41.1% of poll respondents didn't do so.
Key Lesson 1. Life sometimes offers up no-brainers. We should be ready to capitalize.
For example, with only a cursory analysis, we could eliminate 2 out of the 4 poll choices (less than 1000, and more than 1,000,000). But ~41.1% of poll respondents didn't do so.
24/
If the market offers us a pile of oranges of height 100, but prices it at less than the worth of 1,000 oranges, we should load up!
And if the pile is priced higher than 1,000,000 oranges, we should stay away!
Sometimes, it's that obvious.
If the market offers us a pile of oranges of height 100, but prices it at less than the worth of 1,000 oranges, we should load up!
And if the pile is priced higher than 1,000,000 oranges, we should stay away!
Sometimes, it's that obvious.
25/
This is what @MohnishPabrai means when he says he looks for opportunities that "strike him across the head with a two-by-four".
Warren Buffett calls such opportunities "one foot bars that he can step over" (as opposed to 7-foot bars that he has to jump over).
This is what @MohnishPabrai means when he says he looks for opportunities that "strike him across the head with a two-by-four".
Warren Buffett calls such opportunities "one foot bars that he can step over" (as opposed to 7-foot bars that he has to jump over).
26/
Key Lesson 2. We should do our best to understand scale.
Often, it's hard to work out the precise relationship between two variables.
But it may easier to work out how one variable *scales* with respect to the other.
Key Lesson 2. We should do our best to understand scale.
Often, it's hard to work out the precise relationship between two variables.
But it may easier to work out how one variable *scales* with respect to the other.
27/
For example, even without knowing a precise formula for the number of oranges in a pile, we may be able to say that this number *scales* cubically with the height of the pile.
No matter what @sweatystartup says, it's not just tomatoes that scale. Oranges do too.
For example, even without knowing a precise formula for the number of oranges in a pile, we may be able to say that this number *scales* cubically with the height of the pile.
No matter what @sweatystartup says, it's not just tomatoes that scale. Oranges do too.
28/
For example, we may be unable to precisely relate a business's revenues to its profits.
But if we understand its cost structure, we may be able to work out how costs *scale* with revenues.
This can give us insight into the business's operating leverage. (h/t @BrianFeroldi)
For example, we may be unable to precisely relate a business's revenues to its profits.
But if we understand its cost structure, we may be able to work out how costs *scale* with revenues.
This can give us insight into the business's operating leverage. (h/t @BrianFeroldi)
29/
Key Lesson 3. Much of investing boils down to intelligent counting.
Examples include counting assets against liabilities (accounting), estimating future cash flows (valuation), and enumerating the ways an investment can succeed or fail (probabilistic analysis).
Key Lesson 3. Much of investing boils down to intelligent counting.
Examples include counting assets against liabilities (accounting), estimating future cash flows (valuation), and enumerating the ways an investment can succeed or fail (probabilistic analysis).
30/
Familiarizing ourselves with basic math concepts -- areas and volumes, arithmetic and geometric progressions, sums of basic sequences (like Formulas 1 and 2 above), permutations and combinations, etc. -- can go a long way in making us better at counting.
Familiarizing ourselves with basic math concepts -- areas and volumes, arithmetic and geometric progressions, sums of basic sequences (like Formulas 1 and 2 above), permutations and combinations, etc. -- can go a long way in making us better at counting.
31/
Key Lesson 4. We must crawl before we can walk, and we must walk before we can run.
For example, before we can count a big pile of oranges of height 100, it helps to understand how a small pile of height 4 is put together.
Key Lesson 4. We must crawl before we can walk, and we must walk before we can run.
For example, before we can count a big pile of oranges of height 100, it helps to understand how a small pile of height 4 is put together.
32/
Similarly, understanding a simple business can pave our way towards understanding more complicated businesses.
Cultivating a good, "first principles" understanding of fundamental concepts can help us build a strong foundation for more advanced topics. Etc.
Similarly, understanding a simple business can pave our way towards understanding more complicated businesses.
Cultivating a good, "first principles" understanding of fundamental concepts can help us build a strong foundation for more advanced topics. Etc.
33/
I want to thank @QuantaMagazine, @7homaslin and @EricaKlarreich for this thread.
I got the orange counting problem from their book, The Prime Number Conspiracy.
From the chapter about the extraordinary mathematician Manjul Bhargava, who solved it when he was just 8:
I want to thank @QuantaMagazine, @7homaslin and @EricaKlarreich for this thread.
I got the orange counting problem from their book, The Prime Number Conspiracy.
From the chapter about the extraordinary mathematician Manjul Bhargava, who solved it when he was just 8:
34/
If you're still with me, thank you very much!
I hope that counting oranges was an interesting exercise in itself, and that it was able to illuminate some useful principles for more general problem solving.
Please stay safe, and enjoy your weekend!
/End
If you're still with me, thank you very much!
I hope that counting oranges was an interesting exercise in itself, and that it was able to illuminate some useful principles for more general problem solving.
Please stay safe, and enjoy your weekend!
/End
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