First, what is an "usual" number-naming scheme? We take a certain hierarchy, , and map a word for it up to certain n and (not necessarily do it for each ). The restriction is only that the rule always keeps. So, under that, Bowers' scheme is usual:

= googol

= giggol

= gaggol

...

= boogol

= biggol (even though we have no word for , it still works)

= baggol

...

= troogol

## Mixed-arrow notation

Let's define so-called mixed-arrow notation as follows:

- is an expression composed by 's and 's.

Rule 1. (# is empty, no arrows at all)

Rule 2. (b=1, regardless of #)

Rule 3. (# ends at up-arrow)

Rule 4. (# ends at down-arrow)

Examples:

Note that because they both drop to multiplication which is associative.

## Unusual naming scheme

Now we want to form names for this unique notation.

Let be 0 and be 1. Then we can map a binary expansion of each number to the arrow-expression. For example, 5 = 101_{2} = . So we can form the prefix for our names, adopting English numbers. For the suffix, I propose "-arrowal". For a and b we choose "3" as the smallest non-trivial case.

So it forms the names for the numbers as follows:

- Zerarrowal = = 27
- Onarrowal = = 27 (not greater than previous term)
- Twarrowal = = 19683
- Thrarrowal = = 7625597484987
- Fourarrowal =

- Fivarrowal =

- Sixarrowal =

- Sevarrowal = (exactly tritri in Bowers' system)

- Eigarrowal =

To estimate eigarrowal we have to determine how fast grows. Consider it:

. So eigarrowal is about , which is, as we know, much less than sevarrowal = . It turned out that in our naming scheme (n+1)-arrowal can be even lesser than n-arrowal.

- Ninarrowal =

- Tenarrowal =

...

- Twentarrowal

- Hundarrowal

- Thousarrowal (superscripts still can be used).

I see no way to pin down this sequence to a standard ordinal hierarchy .