Just as we can find the half-way point of a linear series of values by taking the arithmetic mean or of an exponential series by taking the geometric mean, we can use the same approach to find the between value of the set of tetrated values. Other than just putting all the hyper operations up by one the only change is that you have to put the numbers in descending order, because for example to find the middle value of 2 (2^^1) and 16 (2^^3) you take the super square root of 16^2 and not 2^16 to get to the correct answer, 4 (2^^2).

What's interesting is that when using more than one value the standard right to left tetration gives us incorrect answers. For instance if we take the numbers, 2, 4 and 16 we should still expect 4 to be the half way point as there is no higher or lower numbers added to diverge the original mean. However, if put the numbers in descending order, put them into an exponent tower and take the super cube root, the function is telling us the average is just over 3, and not the correct answer of 4.

If we do it the "wrong way" however and solve the exponents left to right as (16^4)^2, adjusting the super cube root function to obey the same rules, we get the correct value of 4 and as far as I can tell (please correct me if I'm wrong) it works every time, no matter the sequence.

So my question is, why does left to right tetration work and right to left not, if right to left is supposed to be more mathematically sound? And is there any way you can fix it so right to left tetration averages work?

Sources

http://www.wolframalpha.com/input/?i=%28...5E4%29%5E2

http://www.wolframalpha.com/input/?i=x%5...284%5E2%29

EDIT: Too many typos

What's interesting is that when using more than one value the standard right to left tetration gives us incorrect answers. For instance if we take the numbers, 2, 4 and 16 we should still expect 4 to be the half way point as there is no higher or lower numbers added to diverge the original mean. However, if put the numbers in descending order, put them into an exponent tower and take the super cube root, the function is telling us the average is just over 3, and not the correct answer of 4.

If we do it the "wrong way" however and solve the exponents left to right as (16^4)^2, adjusting the super cube root function to obey the same rules, we get the correct value of 4 and as far as I can tell (please correct me if I'm wrong) it works every time, no matter the sequence.

So my question is, why does left to right tetration work and right to left not, if right to left is supposed to be more mathematically sound? And is there any way you can fix it so right to left tetration averages work?

Sources

http://www.wolframalpha.com/input/?i=%28...5E4%29%5E2

http://www.wolframalpha.com/input/?i=x%5...284%5E2%29

EDIT: Too many typos