Finding The Necessary Amount Of Numbers To Select From A Set

How many numbers must be selected from the set {1,2,3,4,5,6}to guarantee that at least one pair of these

numbers add up to 7?

The answer ends up involving the fact that, since there are only three subsets--namely, {1,6}, {2,5}, and {3,4}--a person is only required to select four numbers from the set. This is due to the fact that at least two of them must fall within the same subset.

I am very frustrated right now (which is probably why I can't think properly, at the moment); and I am becoming more frustrated because the book gives the impression that it is so simple. I don't understand this--not one bit. Can someone please help me?

EDIT: I am having a lot of difficulty, right now. I seem to be questioning every single thing that I do, and sometimes I can just accept an idea, even the ones that appear to be most intuitive. What's wrong with me? How did I get myself into this sort of mindset?

Re: Finding The Necessary Amount Of Numbers To Select From A Set

Hold on...I think I might have figured it out. Just allow me time to type up my solution.

Re: Finding The Necessary Amount Of Numbers To Select From A Set

Let $\displaystyle S = \{1,2,3,4,5,6\}$. The subsets of S, whose elements all summed together result in 7, are: $\displaystyle \{1,6\}$, $\displaystyle \{2,5\}$, $\displaystyle \{3,4\}$.

Since there are only three sets, all of which contain only two elements, and $\displaystyle \{1,6\} \cup \{2,5\} \cup \{3,4\} = S$, by selecting four numbers from S at least two of the elements must fall within the same subset. Although, through experimentation, it would apear that any way of selecting four numbers from S results in exactly two of the three subsets being selected.

Re: Finding The Necessary Amount Of Numbers To Select From A Set

Quote:

Originally Posted by

**Bashyboy** How many numbers must be selected from the set {1,2,3,4,5,6}to guarantee that at least one pair of these numbers add up to 7?

Is it possible to select a subset of three of those that does not have that property ?

Is it possible to select a subset of four of those that does not have that property ?

Re: Finding The Necessary Amount Of Numbers To Select From A Set

Hmm, I want to say the answer to both of those questions is no.

Re: Finding The Necessary Amount Of Numbers To Select From A Set

Quote:

Originally Posted by

**Bashyboy** Hmm, I want to say the answer to both of those questions is no.

Show us a subset of three not having the property.

Show us a subset of four not having the property.

Re: Finding The Necessary Amount Of Numbers To Select From A Set

Hello, Bashyboy!

Plato asked you the proper questions.

*Think* before you answer.

Quote:

Let $\displaystyle S = \{1,2,3,4,5,6\}$.

The pairs whose sum is 7 are: $\displaystyle \{1,6\}$, $\displaystyle \{2,5\}$, $\displaystyle \{3,4\}$.

"Is it possible to select a subset of three of those that does not have that property?"

Yes, we can select one number from each of those three subsets.

. . Examples: .$\displaystyle \{1,2,3\},\:\{1,2,4\},\:\{1,5,3\},\:\{1,5,4\}\: \hdots$

"Is it possible to select a subset of four of those that does *not* have that property?"

No . . . selecting a fourth number *will* give us a sum of 7.