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There is 3 squairres and 5 tricks. show if it can be right.

Max0815  Aug 22, 2018
edited by Max0815  Aug 24, 2018
edited by Max0815  Aug 24, 2018
edited by Max0815  Aug 24, 2018
 #1
avatar+96 
+2

We will proceed by contradiction and induction.

Assume that there is no loop. If two bridges connect the same pair of islands, then we have a loop. So, we cannot have two bridges connecting the same islands.

Case 1: Assume now that each island has two bridges. Denote the islands as . WLOG,  connects to ,  connects to , and so on. However, this yields a contradiction since  and  both have one bridge. To fix this, we can connect them together, which results in the whole graph being a cycle of length . On the other hand, we now have  islands and  bridges. Uh-oh. Let's try to fix this by adding in an arbitrary bridge connecting  and . Now note that we have a cycle connecting  which is a contradiction. Thus, if each island has at least two bridges, then there must be (at least) a cycle.

Case 2: If there is an island with less than \(2\) bridges, then we can "ignore" the island and all bridges extending out of the island. We can keep performing this operation until all islands have at least \(2\) bridges. Now, by Case 1, we are done.

 

Otherwise; answered at https://artofproblemsolving.com/community/c243175h1500558_combinatorics_2_induction

Olpers  Aug 23, 2018
 #2
avatar+174 
+1

um...ok well thanks i guess

Max0815  Aug 24, 2018
edited by Max0815  Aug 24, 2018
 #3
avatar+96 
+2

Haha why did you delete the question?

Olpers  Aug 24, 2018
 #4
avatar+174 
+1

total nonsense xD

Max0815  Aug 24, 2018
edited by Max0815  Aug 24, 2018
 #5
avatar+93299 
0

Max that sounds like total nonsense.

If your teacher did not want other kids to cheat they would not want you to cheat either!

You take if off to attempt to disguise the fact that you are cheating.

 

It is extremely rude. People have been banned from this site for doing this.

 

People go to the effort of answering your question and then you blank your question. You have not even bothered to give Olpers a proper thankyou.

You have not even given him/her a point although I notice you are quite happy to give yourself a point just for asking a question.

It is not good that you have drawn attention to yourself in this way.

 

Do not ever delete any of your question in the future!

Melody  Aug 24, 2018
 #7
avatar+96 
+1

Haha I didn't really answer the question, but yeah I agree with you.

Olpers  Aug 24, 2018
 #6
avatar+174 
+1

I am sorry melody. I will not delete my questions in the future.

Max0815  Aug 24, 2018
 #8
avatar+93299 
+1

Good, thank you :)

Melody  Aug 24, 2018

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