Fumbling in the dark

According to the queston there are $13$ pairs of shoes in the closet which means that there are 26 shoes in it. Calvin wants to get a matching pair.If Calvin extracts $13$ shoes then all of them might be of different pairs.But if he takes out one more shoe then he is sure to get at least one matching pair. Thus the minimum number of shoes he must remove from the closet to get a pair of matching shoes is $13+1=\boxed{14}$

From the pigeonhole principle , we know that $\boxed{14}$ shoes are needed to guarantee that two are a matching pair.

Not yet convinced?

• 13 shoes aren't enough, because they might turn out to be all distinct.
• 14 shoes are enough, because if the first 13 are distinct, the 14th will match one of the first 13.

So 14 is the minimum.

13 pairs of shoes in his closet, so there are 26 shoes in the closet. in order to get minimum a pair shoes, so we must take 14 shoes.

In order to $\textit{guarantee}$ a matching pair, we must consider the worst case scenario: After blindly removing $N$ shoes, no matter what shoe Calvin removes, he has removed two of the same shoe. If this worst case scenario occurs, Calvin will have removed $13$ shoes, and the next shoe he removes will be the $\boxed{14}$ th shoe.

So let's say that Calvin is very unlucky so he keeps on getting shoes that he hasn't found the pair of yet. So he takes 1,2,3,4,5,6,7,8,9,10,11,12,13 shoes, and by now he's gotten one shoe of every pair. So, the next shoe he gets will definitely be a match of one of the shoes he's already picked, so the 14th shoe guarantees his matching pair.

Worst case is when you pick 13 shoes and all of them are from diferent pairs.

once you've reached this stage you know that the next shoe you pick it is going to be one kind that you have already picked.

Then, 14 shoes is the minimum.

if we observe that we have 13 boxes and 14 balls then atleast one box has two balls....similarly.....simple PHP!!!

We look at the worse case scenario: He takes 13 shoes of different pairs. Now he only needs to take one more shoe to guarantee a pair. 13+1=14.

He has 13 pairs, i.e. 26 shoes. If he takes 14 shoes out of them then even if he has all 13 left-foot shoes, then he definitely will have a right-foot shoe, as there are 13 left-foot shoes and 13 right-foot shoes. :)

The worst case scenario is if Calvin chooses one shoe from every pair (how unlucky!). From this point forward, if he picks up another shoe, Calvin must have a pair. $13+1=\boxed{14}$ .

If he has 13 pairs, then the worst case he can have is grabbing one shoe from every pair before pulling out the last one that matches. Therefore, he picks 13 that don't match, and then the 14th is the match.

If there are 13 pairs of shoes then there are 26 shoes.So he must remove 14 shoes to have a right and a left shoe.

Suppose that shoes left with $L_i$ and shoes right with $R_i$ , $i=1,2,...,13$

First we choice 13 shoes

$L_i$ or $R_i$

For example: $(L_1,L_2,...,L_13$

For every index $i$ If we choice one shoe again (Suppose that $R_3$ for the example )

We will get a mathching pair shoes. ( $L_3,R_3$ )

So, The answer are $\fbox{14}$

If Calvin draws 14 shoes, by the Pigeonhole Principle, there exists at least a matching pair of shoes, hence if he removes 14 shoes he is guaranteed to have a matching pair. Note that 14 is the smallest number, as any number less than 14, e.g. 13, will not guarantee Calvin a matching pair as he might draw exactly one shoe from the 13 different pairs.

For each type of shoe, create 13 boxes. If 14 shoes are chosen and placed in their correct box, then at least one box must contain 2 shoes. Therefore the smallest number of shoes required is no greater than 14. To show that 14 is the minimum number, you'd then have to show why 13 may not be enough. This is easy: you might be unfortunate enough to pick one shoe of each of the 13 types before striking a pair.

With the word "guarantee", we can only assume the worst possible luck. The unluckiest thing to happen is getting a continuous streak of 13 shoes from different pairs. But anyway, the 14th drawing will always turn the tables for us, by matching to one of the previously distinct 13.

Exactly their are 26 shoes (13 pairs).they are asking "the minimum number of shoes he must pick in order to guarantee that he has a matching pair among them". So we can't guarantee that first pair comes when he picks 9 or 10 or 12th shoe , Instead we can divide sum of shoes by 2 . 26/2=13. in assumption we take first 13 shoes are all unique without their pair then after 14th shoe guaranteed that his pair has found before itself

The worst-case scenario is that Calvin takes $13$ of, say, left shoes. However, the next shoe would then always be a right shoe, and that shoe will complete a pair in the left shoes already picked. Thus, $\boxed{14}$ .

I think answer should be 27 single shoes to guarantee having matching pair

I know there must be 14 shoes left in order to guarantee there's at least one matching pair. The question is "What is the minimum number of shoes he must remove in order to guarantee that he has a matching pair?". I think the question is a little bit confusing.
* If Calvin chooses the shoes from the closet, then he needs to remove at least 12 to guarantee there will be at least one matching pair among the 14 left. * But if he chooses the shoes from the ones removed from the closet then he needs to remove at least 14 to guarantee there will be at least one matching pair among such 14 shoes removed from the closet.

Calvin has $13$ pairs of shoes, which means he has $26$ shoes.

Worst case scenario: Calvin extracts $13$ different shoes.

Therefore, adding exactly $1$ shoe will guarantee that he'll have a matching pair.

Thus the minimum # of shoes he must extract is $13+1=\boxed{14}$ shoes according to the pigeonhole principle.

because there are for example objects will be 14 that you will have