In this article we will cover chained arrow notation, a notation for expressing certain very large numbers, which was invented by John Conway, hence the name Conway chained arrow notation.
3-length chains and 4-length chains
We will start by covering chains with 3 numbers. First off, here are the rules for this notation:
a -> b = ab
a -> 1 -> b = a (In general, any chain with a 1 is equal to the value of the same chain with the 1 and everything to the right of it removed)
a -> b -> c = a^c b
a -> b -> c -> d = a -> b -> (a -> b -> c-1 -> d) -> d-1
We will begin by evaluating a chain that evaluates to a relatively small number. For example:
2 -> 3 -> 3
= 2 -> (2 -> 2 -> 3) -> 2
= 2 -> (2 -> (2 -> 1 -> 3) -> 2) -> 2
= 2 -> (2 -> 2 -> 2) -> 2
= 2 -> (2 -> (2 -> 1 -> 2) -> 1) -> 2
= 2 -> (2 -> 2 -> 1) -> 2
= 2 -> (2 -> 2) -> 2
= 2 -> 22 -> 2
= 2 -> 4 -> 2
= 2 -> (2 -> 3 -> 2) -> 1
= 2 -> (2 -> (2 -> 2 -> 2) -> 1) -> 1
= 2 -> (2 -> (2 -> (2 -> 1 -> 2) -> 1) -> 1) -> 1
= 2 -> (2 -> (2 -> 2 -> 1) -> 1) -> 1)
= 2 -> (2 -> (2 -> 2)) (remove the 1s)
= 2 -> (2 -> 4))
= 2 -> 24
= 2 -> 16
= 216
= 65,536
It can be shown that any chain beginning with two 2s will degenerate to 4, because it is the same as an expression of the form 2^c 2, which will always equal 4 regardless of the number of arrows. a -> b -> c is equal to a^^...^^b where the number of up-arrows is c, so we can know that the chain 3 -> 3-> 2 will solve to 3^^3 which is 7625597484987, and the chain 5 -> 4 -> 3 will solve to 5^^^4. Now we will examine 4-length chains. We will start by solving the chain 3 -> 2 -> 2 -> 2.
3 -> 2 -> 2 -> 2
= 3 -> 2 -> (3 -> 2 -> 1 -> 2) -> 1
= 3 -> 2 -> (3 -> 2) (drop the 1s)
= 3 -> 2 -> 9
= 3 -> 3 -> 8
When fully expanded, this will evaluate to 3 -> (3 -> (3 -> (3 -> (3 -> (3 -> (3 -> 3 -> 1) -> 2) -> 3) -> 4) -> 5) -> 6) -> 7. Its value is 3^^^^^^^^3.
3 -> 3 -> 3 -> 3
= 3 -> 3 -> (3 -> 3 -> 2 -> 3) -> 2
= 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 1 -> 3) -> 2) -> 2
= 3 -> 3 -> (3 -> 3 -> 27 -> 2) -> 2)
= 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 26 -> 2) -> 1) -> 2)
= 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 25 -> 2) -> 1) -> 1) -> 2
= 3 -> 3 -> N -> 2
where N is 3^^^^^^^^^^^^^^^^^^^^^^...^^^^^^^^^^^^^^^^^^^^^^^^3 where the number of ^s is 3^^^^^^^^^...^^^^^^^^^3 where the number of ^s is 3^^^^^^...^^^^^^3 ... ... where the number of ^s is 3^^^^^^^^^^^^^^...^^^^^^^^^^3 where the number of ^s is 3^^^^^^^^^^^^^^^^^^^^^^^^^^^3 where the number of ^s is 27, where the number of steps is 27
Yikes! With merely a chain of four 3s, we have left Graham's number in the dust!
4 -> 4 -> 4 -> 4, the fourth of the "Conway numbers" is the largest number mentioned in Conway's book.
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