It begins with mathematicians coloring the edges of n-dimensional hypercubes. Why would they do such a thing? I don't know. Maybe they wanted to gussy them up a bit.
Anyway, the mathematicians wanted to know how many dimensions your cube would have to be in order to guarantee that a certain method of coloring its edges would contain at least one example of a particular result. If you want to understand what the method is - or what result they're looking for - you'll have to ask someone else. I'm a long way from understanding that part of it.
So how many dimensions does the cube have to be? The mathematicians don't know exactly. But they've narrowed it down. It must be at least 13. But it won't be any larger than Graham's number. [As in, Ronald Graham. He's the guy who invented the "Erdos number".]
Then what is Graham's number? Before you can talk about Graham's number, you have to understand Knuth's up-arrow notation. [As in Donald Knuth. He's the guy who created the TeX typesetting system.]
Now, regular exponents are often represented by a single up-arrow.
A^B = A*A*A...*A with there being B copies of A.
3^2 = 3*3 = 9
3^3 = 3*3*3 = 27
3^4 = 3*3*3*3 = 81
What Knuth did was define a system using additional arrows.
A^^B = A^(A^(...^A) with there being B copies of A
[With exponents, you always work from right to left.]
In other words, A raised to the power of itself in a tower of exponents with B levels.
3^^2 = 3 to the power of 3 = 27
3^^3 = 3 to the power of (3 to the power of 3) = 3 to the power of 27 = 7,625,597,484,987
That's 7.6 trillion - which is nothing to sneeze at. But it's visualizable. If you took 3 Sears Towers and filled them with pennies, that's more or less the number of pennies you'd have.
3^^4 = 3 to the power of (3 to the power of (3 to the power of 3)) = 3 to the power of 7,625,597,484,987
Here we have gone beyond the visualizable. By comparison, the number of Planck volumes - the smallest volume in which the known laws of physics make any sense - contained in the entire known universe is 10 to the power of 185. So with 3 to the power of 7,625,597,484,987, we can safely say that we have entered the realm of Big.
Now let's add a third arrow:
A^^^B = A^^(A^^(...^^A)) with there being B copies of A
[As with exponents, when using arrows, you work from right to left.]
Note how, as before, the number of arrows between terms on the right side of the equal-sign is one less than the number of arrows between terms on the left side of the equal-sign.
3^^^2 = 3^^3 = 7,625,597,484,987
3^^^3 = 3^^(3^^3) = 3^^7,625,597,484,987
In other words: an exponent tower of 3's that is stacked 7,625,597,484,987 levels high.
To give some idea of this: if it took you a second to calculate each new level of the stack, it would take more than 240,000 years to finish. Now we are in the realm of Stupid Big.
Let us call this number X.
3^^^4 = 3^^(3^^(3^^3)) = 3^^X = An exponent tower of 3's that is X levels high.
3^^^5 = An exponent tower of 3's that is 3^^^4 levels high.
3^^^6 = An exponent tower of 3's that is 3^^^5 levels high
Continue this series until the number after the three arrows is X.
This last number equals 3^^^^X. Let us call this Insanely Big number G1. And remember that it took only four arrows to get there.
Now things get interesting.
G2 = 3^^...^^3 with there being G1 arrows.
G3 = 3^^^...^^^3 with there being G2 arrows.
Continue this series until you reach G64.
THAT is Graham's number.
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