When any two base 3 numbers (without the digit 2 in them) are averaged, why does the average (if it is an integer) always have a digit 2 in it?
When any two base 3 numbers (without the digit 2 in them) are averaged, why does the average (if it is an integer) always have a digit 2 in it?
I'm sorta out of practice with my base 3, but here goes.
The problem asks why the average always has a 2 in it. I think the assumption is not true.
All I need to do to disprove a generality is just find one exception.
Take any number in base three. Say that it does not have a 2 in it.
Take a second number, make this number the same number as the first.
Average them and what you get is the original number, which did not contain a 2.
No, it's not cheating, because the problem said ANY two base 3 numbers.
If it had wanted any two different base 3 numbers, it should have said that,
but I think that could be disproven, too, although it would be less intuitive.
