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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?

 Jul 10, 2022
 #1
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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. 

 Jul 11, 2022
 #2
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Ok, thanks you for your help!

Guest Jul 11, 2022

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