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Number Difference and Erasure Puzzle (Posted on 2015-09-14) Difficulty: 2 of 5
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?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Too many to list | Comment 1 of 2
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
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