The numbers 1, 3, 3², 3³,...,3¹⁰ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a nonnegative integer.)
After this procedure has been repeated ten times, only a single number will remain.
What values could this number have?

Reverse the order of the numbers to make the subtraction normal.
There are 10! possible order of erasures correspond to various ways of inserting 9 sets of parentheses.  But many of these are repeats.

Once simplified to remove the parentheses some of the subtractions become additions, except for the very first: 3^9 will always be subtracted (and 3^10 will always be positive.)

It turns out that for the other 9 that every possible combination is possible in at least one way. 

The powers of 3 are far enough apart that no two combinations will give the same value.

So there are 2^9=512 possible values for this number ranging from a low of
--------- 29525
to a high of
-+++++++++ 49207

Posted by Jer on 2015-09-16 09:34:03