If I ask you what is the largest number that exists, you might think that there cannot be anything like ‘the largest number’. You can just add 1 to the largest number and then you have an even larger number. A better way of asking this question would be: what is the largest number that has ever been used in mathematics, constructively?
Just to have some fun, I would like you to think of the largest possible number you can think of. You don’t have to know the exact digits; just a rough approximation of a very large number. Now let’s see if it comes close to Graham’s Number. But before that read the following problem.
Imagine a three dimensional cube to start with. Now connect each of the vertices with all the other vertices like in the image shown below with any combination of red and blue lines.
Depending on your colouring scheme, there might be a set of four co-planer points, which are connected by lines of the same colour. e.g. in the above diagram you can see a set of four points exist which are connected by lines of the same colour. For a three dimensional cube, every possible way of colouring these lines will not contain four coplanar points connected by lines of the same colour, as shown below:
In the above example, it is quite easy to see that if we change the colour of the line joining the two bottom right vertices to blue then no set of co-planer points exist.
Or, maybe, we can just use an analogy to understand the question better.
We replace vertices with people who are attending a party. It is a special party and only the people who wear a blue or a red dress are allowed to get in. These people are randomly sitting at various tables which are distributed across the hallway. What the problem is really asking is how many people need to be there so at least at one of the tables there would be people who are dressed exactly alike and are sitting in the opposite corners.
The answer to this question lies somewhere between 13 and Graham’s number.
The proof of this question is beyond the scope of this article but we can still try to wrap our head around this big number and try to see how BIG it really is.
The easiest way of writing down this number is using Knuth’s arrow notation:
Although this number may already be beyond comprehension, it is barely the beginning.
The next step is to add another arrow:
And call this number G1:
This is the part where things blow really out of proportion.
G1 is the number of arrows between the two 3s in the next number, which we will call G2.
At this step, I would like you to realise what we have done here. Adding just one arrow to 3↑3 which is 27, ballooned it to 7.6 trillion. And the number of arrows in G2 is the number which itself is unimaginably huge.
G3 is equal to 3↑↑↑↑………↑↑↑↑↑3 , where the number of arrows is G2.
We keep repeating this step till we define G64 to be 3↑↑↑↑………↑↑↑↑↑3, where the number of arrows is G63.
This number is so big that the number of digits in this number is larger than the number of numbers we would need to fill the entire universe.
If you take all the matter in the universe and use it to make one big memory bank which stores information using Qubits, then even a Googolplex of these memory banks cannot store this number.
In fact, it’s so big that we have no way of calculating what the number really is; the smallest possible way of writing down this number is using Knuth’s arrow notation as was mentioned previously. But we do know the last 500 digits and the last digit is 7.
So, was the number you first thought of bigger than Graham’s Number?
Upon writing this article I came across the thought that not only is Graham’s Number truly big but when you write it down using Knuth’s arrow notation, it contains the largest amount of information that can be written down using a pen. In other words, Knuth’s arrow notation for Graham’s number contains more information than everything ever written by humans in all of history collectively.