Welcome to the staging ground for new communities! Each proposal has a description in the "Descriptions" category and a body of questions and answers in "Incubator Q&A". You can ask questions (and get answers, we hope!) right away, and start new proposals.
Are you here to participate in a specific proposal? Click on the proposal tag (with the dark outline) to see only posts about that proposal and not all of the others that are in progress. Tags are at the bottom of each post.
Starting with 2014 "+" signs and 2015 "−" signs, you delete signs until one remains. What’s left? Question
A large whiteboard has 2014 "+" signs and 2015 "−" signs written on it. You are allowed to delete two of the symbols and replace them according to the following two rules.
If the two deleted symbols are the same, then replace them by "+".
If the two deleted symbols are different, then replace them by "−".
You repeat this until there is only one symbol left.
Which symbol is it? Justify your answer.
Attribution: UK Mathematical Olympiad for Girls 2014
2 answers
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| will.octagon.gibson | (no comment) | Aug 3, 2026 at 16:54 |
Solution
-Reasoning
Two of the same sign collapse to +; two different signs collapse to -. This is isomorphic to the multiplication rules, so the entire question is isomorphic to:A large whiteboard has 2014 (+1) and 2015 (−1) written on it. You are allowed to replace two numbers by their product. Repeat until there is only one number left.
Alternatively, and more formally: the replacement rules maintain the parity of the number of - signs. Since that number is initially odd, when only one sign remains it must be -.
0 comment threads
Withdrawn -- this is incorrect.
answer
You can produce whichever outcome you want.
method 1
Start by collapsing most of this away.
Delete 2014 "-" in pairs. We now have 3021 "+" and 1 "-".
Delete 1 "-" and 1 "+". We now have 3020 "+" and 2 "-".
Delete the two "-". We now have 3021 "+" and 0 "-".
Everything is now "+" and deleting two of the same results in a "+". There's no way for "-" to re-enter, so if we keep doing this, we'll eventually end up with 1 "+".
method 2
What happens if we start by deleting (pairwise) 2014 of each? We end up with 0 "+" and 2015 "-".
We delete 2014 "-" pairwise, ending up with 1007 "+" and 1 "-".
We can obviously delete the "-" and one "+" and reduce to the previous method. Instead, let's collapse the "+"s. "++" reduces to "+" and we can keep doing that until we're down to 1 "+" and 1 "-". Deleting them results in a "-".
0 comment threads