Can You Really Solve The Staircase Paradox?

The staircase paradox is one of those challenges that deeply questions our fundamental logic. It is a geometrical problem that has occurred in varying forms over the years in mathematical circles. The complex versions involve the concepts of circles and squares, whereas the simpler versions involve the concepts of squares and triangles. I will be explaining the paradox using a simple version in this article.

As the reader, no profound mathematical knowledge is required from your side to understand the explanation in this article. However, don’t let the simplicity of the problem fool you. This simple paradox poses a profound challenge we face as human beings.

By the end of this article, I bet that you will have deep questions about the business of knowledge and logic. On my part, I will try my best to answer the question of whether one can solve this paradox. Let’s start with a short definition of the problem we are trying to solve here.

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What is the Total Length of The Stairs?

Imagine that we have a staircase whose base is 4 Centimetres (Cm) long and height is 3 Cm long. As for the stairs themselves, we have two of them, each half as long and half as high as the entire staircase.

Now imagine that we wrap a measuring tape around the stairs to measure the total distance covered by the stairs. In short, we are interested in measuring the length of a string that would run all along the outline of the stairs (bottom to top or vice-versa). This looks like an extremely simple problem, right?

We have two stairs, each half as high and half as long as the entire staircase. We could split these into lengthwise components and height-wise components and just add them up. Finally, the sum would give us the distance covered by the stairs (henceforth refererred to as “total length of the stairs”).

Total length of stairs = (Sum of lengthwise components) + (Sum of height-wise components) = (2 + 2) + (1.5 + 1.5) = 7 Cm.

What we have essentially figured out here is that the total length of the stairs is equal to the sum of the length and height of the staircase. If you think that this is too easy, be warned that we are just getting started here.

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What Happens When the Number of Stairs Keeps Doubling?

Let’s say that we are asking the same question as before, but we choose to double the number of stairs. So, the second time around, we have 4 stairs. Each of these is one-fourth as long and one-fourth as high as the entire staircase.

In order to find the total length of the stairs, we could employ the same strategy as before, and we would end up with the following:

Total length of stairs = (Sum of lengthwise components) + (Sum of height-wise components) = (1 + 1 + 1 + 1) + (0.75 + 0.75 + 0.75 + 0.75) = 7 Cm.

The key point to take away here is that doubling the number of stairs did not cause any change to the total length of the stairs from before. The total length is still equal to 7 Cm.

What if we choose to continue the doubling process? Let’s do just that, and see where this goes.

The first time around, we end up with 8 stairs. We then double this to get 16 stairs. The right-most figure has 32 stairs. In each of these cases, we can still employ the same logic of summing up the lengthwise components and height-wise components to get the total length of the stairs.

By doubling the number of stairs, we are dividing the lengthwise and height-wise components by 2 each time around. This does not change the total length. If we sum them up, we will still end up with a total stair length of 7 Cm.

But if you were curious enough, you would have noticed something interesting about the rightmost figure. It resembles a triangle. In short, the more the number of stairs, the closer the staircase starts looking like a triangle. This brings us to the centre of our discussion in this article.

The Staircase Paradox

Let’s say that we keep doubling the number of stairs to the point that the stairs converge into a line. This line would then complete the triangle. The staircase has now become a triangle. If you are preceptive, you will notice that this is a right-angled triangle.

This enables us to use the Pythagorean theorem, which states that the hypotenuse of a right angled triangle is the square root of the sum of the squares of the other two sides. In our case, the total length of the stairs now become the hypotenuse. Therefore:

Total length of the stairs = √(3² + 4²) = √(9 + 16) = √25 = 5 Cm

We have now arrived at a very confusing result. Until this point, we were certain that the total length of the stairs was equal to 7 Cm. How can 7 Cm be equal to 5 Cm?

This is essentially the staircase paradox!

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How to Understand the Staircase Paradox?

“A paradox is a logically self-contradictory statement or a statement that runs contrary to one’s expectation.” 

— Wolfram Mathworld

“It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.” 

— Oxford Dictionary

The first and the most important step is to appreciate the fact that we followed a sound and logical approach to arrive at the point where the paradox occurred. If there were no mistakes in our logic, what else could have caused this?

We are essentially trying to measure a continuous entity (a line) initially approximated by discrete entities (individual stairs). Then, we try to extrapolate our logic at a limit where the approximation is no longer an approximation. For the approximation to no longer be an approximation, we would need an infinite number of stairs. Infinity is not a number. And things start getting really weird around the concept of infinity. For reference, I have written an article that explains why infinite infinities exist.

In short, even though our logic so far is correct, it is not sufficient to avoid the staircase paradox. There are unfortunately no two ways about this conclusion. This paradox, by definition, is unsolvable. Therefore, the only hope we have is to eliminate it from our question or avoid it, if we can.

How to Eliminate the Staircase Paradox?

I have come across a few approaches that treat the staircase paradox using the concept of fractals. However, I am not convinced that this is correct. In one of my previous articles, I explained how measuring any coastline is impossible because of its fractal nature. Coastlines are natural phenomena that feature fractality, whereas we constructed the mathematically perfect stairs in this article. They do not have a fractal nature.

Since we are dealing with infinity here, the only way to eliminate the staircase paradox (or prove that there is no fractality occurring here) is to use advanced mathematics. I am unfortunately yet to come across a simpler solution.

As I intend this article to be accessible for anyone without a mathematical background, it is counterproductive to compress advanced mathematics into it. I will cover the paradox elimination procedure (the mathematical treatment of the problem) in a separate future article.

For now, it suffices if you are aware that it is possible to eliminate this paradox. The only way to do it, however, is to turn to advanced mathematics.

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Further reading that might interest you:  Logarithms: The Long Forgotten Story of Scientific Progress, Why Earning More Leads To Lesser Satisfaction, and Why Do You See Mirrors Flipping Words?