Question 1204579: Peter has many socks, all the same except that they are in nine different colours. He is leaving to catch an early train to go on a business trip, and he does not want to wake his wife, so he must pack in the dark. He needs six pairs of socks, each sock in each pair the same colour. How many socks must he take from his drawer to be sure of achieving this?
Found 4 solutions by greenestamps, ikleyn, Edwin McCravy, mccravyedwin:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
The statement of the problem makes it impossible to answer the question.
If in fact there are exactly nine pairs of socks of different colors, then the worst case is that he first picks one sock of each of the nine colors; then each of the next six socks he picks will make a matching pair. That makes 9+6 = 15 socks to guarantee that he will have six pairs.
(one possible) ANSWER: 15
But the problem only says that there are "many" socks.
Consider the absurd case in which there are 8 pairs of socks of different colors and 100 socks of a ninth color. Then the worst case would be that he again first picks one sock of each of the nine colors. Then the tenth sock he picks will make the first matching pair. But after that the worst case would be that he then picks only socks of the last color. He needs five more matching pairs, which means he would have to pick ten more socks of that color. In that case, the number of socks he would have to pick to guarantee having six matching pairs is 9+1+10 = 20.
Upon first reading the problem, I was immediately suspicious of the "many" socks, suspecting that that wording would make it impossible to answer the question -- and indeed that was true.
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Tutor @ikleyn jumps at every chance she can find to disparage responses from other tutors. But often her remarks indicate she is not reading those responses.
As my response states, 15 socks are enough to guarantee 6 matching pairs IF THERE ARE EXACTLY 9 MATCHING PAIRS TO CHOOSE FROM.
And, as she states in her "corrected" response, 20 socks must be chosen if there are more socks than just 9 matching pairs -- which is exactly what my response says.
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To tutor @ikleyn....
What a lame response to my post about your comment regarding my post. You say you have an absolute ear for math; too bad your English is not very good, and/or that you can't read.
My response clearly stated that the answer was 20 if we don't know how many total socks are in the drawer, and that the answer was 15 if there were exactly 9 pairs of matching socks.
Your suggestion that there was anything wrong with my response is absolutely unfounded.
Grow up and stop feeling like you need to show you are better than everyone else.
Answer by ikleyn(52824)  (Show Source):
You can put this solution on YOUR website! .
Peter has many socks, all the same except that they are in nine different colours.
He is leaving to catch an early train to go on a business trip,
and he does not want to wake his wife, so he must pack in the dark.
He needs six pairs of socks, each sock in each pair the same colour.
How many socks must he take from his drawer to be sure of achieving this?
~~~~~~~~~~~~~~~~~~~~~
The problem's formulation in the post is correct and allows to get an accurate solution.
The term "many socks", used in the formulation, means
" many enough to make all necessary manipulations providing minimum guaranteed amount. "
Solution
In the worst case, Peter takes 9 socks of different colors (let say, of the colors 1,2,3,4,5,6,7,8,9),
from which he can not get a pair of the same color.
But when he takes any next sock (n. 10) from the great collection,
he inevitably will get one pair of the same color. Let this "same color" is "color 9".
Ok. So, after taking 10 socks, Peter has at least one pair; in the worst case, exactly
one pair. I will refer to it as "pair 1".
Next step Peter takes some sock n.11 from the great collection.
In the worst case it is a sock of the color 9 - then Peter just has the same "pair 1"
and 9 socks of different colors; but taking next sock n.12, he inevitably will have
the second matching pair.
Thus, after taking 12 socks from the great collection, in worst case Peter has 2 matching pairs
and (or plus) 8 non-matching socks.
You see this repeating cycle in my reasoning / (in the procedure):
- in the worst case, there are 8 non-matching socks in the selected collection;
- then adding two socks from the great collection makes/(adds) one additional
matching pair in/(to) the selected collection, in the worst case.
From it, easy logic allows us to conclude that the selected collection must have
8 + 2*6 = 8 + 12 = 20 socks (in the worst case) to provide 6 matching pairs.
ANSWER. Taking 20 soscks without looking at them guarantees 6 pairs of matching socks.
Solved.
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15 taken socks, as stated in the post by @greenestamps, do not guarantee 6 matching socks.
20 taken socks, selected randomly without looking, do guarantee 6 matching socks.
So, having 20+8 = 28 socks of 9 different colors in great collection, as described in the problem,
is ENOUGH and does guarantee that the solution is possible.
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To @greenestamps:
Some people have absolute ear for music: when music plays, they can recognize and name each note.
My daughter has such earing for music: when a piano plays, she sees the notes running in front of her eyes.
Similarly, I have absolute ear for Math, in that its part which I know.
It means that when I hear or see Math spoken or written, I 100% can say, where it is wrong
or how it should sound or how it should be written in a right way.
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website!
I misread and thought there were 6 colors and said the answer is 17. Here was my
1st answer, which would have been correct if there had only been 6 colors. The
correction is posted below. It's the same way to look at it, it just changes the
numbers.
The worst case is that he picks his first 6 pairs all of different colors. Then
he keeps picking socks of the same color as the 7th sock, and the sock it
matched, until he has 12 of that same color and only 1 sock of each of the other 5
colors. So he must pick 17 socks to be absolutely sure that he has 6 pairs of
matching socks.
Edwin
Answer by mccravyedwin(408)  (Show Source):
You can put this solution on YOUR website!
Sorry I misread and thought there were 6 colors, not 9. Here is the correction:
The worst case is that he picks his first 9 pairs all of different colors. Then
he keeps picking socks of the same color as the 10th sock, and the sock it
matched, until he has 12 of that same color and only 1 sock of each of the other
8 colors. So he must pick 20 socks to be absolutely sure that he has 6 pairs of
matching socks.
OK, I agree with Ikleyn, it's 20.
[The way I worded it, it sounds as though he had to pick those 20 socks in some
particular order, but he could possibly pick the same 20 socks in any order.]
Edwin
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