How to have better discrete math insight

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How to have better discrete math "insight"

Greetings:


I came a cross a textbook example in a discrete math book that I have been reading on my own, and I thought this example in the book was a good example of what I want to be good at:


Given integers from 0-9 arranged in a circle, is it true that at least one group where a group is 3 consecutive integer around the circle has a sum that is at least 14?





the solution is that using proof by contradiction, if the statement is not true then we have a sum of the total groups equals 130 at most, but then we must regconize that the sum of all groups is 3 times the total sum of the entire circle. (geo sum of 1 to 9) which is 135, hence contradiction.




my question is, suppose one have not seen this problem, how would one go about this? how would one be able to "see" that total sum of groups is just 3 times the overall circle sum? I read the solution immediately after the question so I guess I ruin the chance for myself to figure out, but I'm asking generally, for discrete math, counting, probability, what are the "healthy" thought process or creativity that would enable one to be better at devising creative and elegant solutions to these types of problems?\


thanks
 

Answers and Replies

  • #2
673
314
Greetings:


I came a cross a textbook example in a discrete math book that I have been reading on my own, and I thought this example in the book was a good example of what I want to be good at:


Given integers from 0-9 arranged in a circle, is it true that at least one group where a group is 3 consecutive integer around the circle has a sum that is at least 14?





the solution is that using proof by contradiction, if the statement is not true then we have a sum of the total groups equals 130 at most, but then we must regconize that the sum of all groups is 3 times the total sum of the entire circle. (geo sum of 1 to 9) which is 135, hence contradiction.




my question is, suppose one have not seen this problem, how would one go about this? how would one be able to "see" that total sum of groups is just 3 times the overall circle sum? I read the solution immediately after the question so I guess I ruin the chance for myself to figure out, but I'm asking generally, for discrete math, counting, probability, what are the "healthy" thought process or creativity that would enable one to be better at devising creative and elegant solutions to these types of problems?\


thanks
I'm not that good at math either. You might try Polya's book how "How to Solve it" or some title like that.
 
  • #3
Stephen Tashi
Science Advisor
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Greetings:


I came a cross a textbook example in a discrete math book

What was the mathematical topic being treated in the chapter where you found this problem? Finding a problem in a textbook is different than finding a problem in a book of miscellaneous puzzles.
 

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