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A lady has some fine gloves and hats in her closet

Sat Aug 21, 2010 12:59 pm  by tartle

A lady has some fine gloves and hats in her closet 14 blue,25 red and 45 yellow. The lights are out and it is totally dark. In spite of darkness, she can make out difference between a hat and glove. She takes out of the closet only if she is sure that it is a glove. How many gloves must she take out to make sure that she has a pair of each color?

A) 72
Mon Jan 03, 2011 1:24 pm  by s.b.

14 blue 25red etc...does it include the no of hats????????? or is it just the no of gloves??
Sun May 29, 2011 7:47 pm  by bds021

worse case scenario all of the red and yellow items are gloves, then she could take out 70 gloves and they could all be either yellow or red. The next two gloves would have to be blue because everything left in the closet is blue.

Of course this is all assuming there exists a pair of gloves for each color
Wed Jul 06, 2011 8:12 pm  by cat

A lady has some fine gloves and hats in her closet: 14 blue, 25 red, and 45 yellow. We assume these numbers are the totals of hats and gloves of each color, and that all gloves come in pairs (one left, one right).

In the worst case, each odd total reflects just one hat, and the even total reflects no hats. Then there are 22 yellow pairs, 12 red pairs, and 7 blue pairs of gloves. She must draw 22+12+1=35 left-handed gloves, and similarly, 35 right-handed gloves.

This is 70 gloves in all, but not just any 70 gloves. She can try each glove she draws to see which hand it fits, and add it to the total for that hand. Even if the gloves were fastened together in pairs, she would still need to draw 35 pairs, which is 70 gloves.
Tue Mar 31, 2015 11:26 am  by quiz-master-India

Thu Jul 14, 2016 3:00 pm  by bathfilms

I reckon the maximum answer is 75 gloves

Remember from the question: "... that ALL gloves come in PAIRS (one left, one right). "

So, if there are 45 yellow items, the maximum number of yellow gloves is 44 plus 1yellow hat
And if there are 25 red items, the maximum number of red gloves is 24 + 1 red hat
If there are hats of all 3 colours, then the maximum number of blue gloves is 12 + 2 blue hats

Worst case scenario (i.e. maximum number of gloves removed before a pair of each is guaranteed):

She could take out 44 yellow gloves, then 24 red gloves = 68 to guarantee

Then she must take out 7 blue to guarantee a blue pair

Total = 68 + 7 = 75

Assuming that the question is not flawed, and given that 75 is not a given answer, then the person setting the question knows that there must be more hats. To maintain PAIRS of gloves (as stated there are), then the increase in hats must be even too.

To reduce the answer from 75 to as low as thirty-something, we need to optimise the reduction in gloves and increase in hats:

Blue gloves = 2, blue hats = 12
Red gloves = 2, red hats = 23
Yellow gloves = 30 OR 28 OR 34

This would mean, glove removal would be:
30Y + 2B + 2R = 34
28 + 2B + 2R = 32
34 + 2B + 2R = 38
As none of these answers are given, we must turn to the only answer that is an even number...

Blue gloves = 4
Red gloves = 24
Yellow gloves = 44

This could mean, glove removal would be:
44Y + 4B + 24R = 72

This is one of several possible answers, but the only possible one given.
It would also mean that there would be 12 hats: 10 blue, 1 red and 1 yellow.
This would give Answer = 72 gloves would need to be removed.
(But not for the reasons given by others)

However, in reality only 71 would need to be removed:
44Y + 24R + 3B = 71

Hence, I propose that all 4 given answers are invalid.

Therefore I will stick with my original value of 75.
Fri Jul 15, 2016 2:06 am  by Dzallen

If we assume there can be any number of R B or Y hats, and left and right gloves are separated, then all four answers are possible:

no. gloves: Y R B
44 24 6 =72 gloves needed
30 10 8 =35 gloves needed
I think it's most reasonable to assume all 12,25 and 45 are numbers of gloves, and that there is no difference between left and right gloves, as this means the answer can only be 72.

Last edited by Dzallen on Mon Jul 18, 2016 1:38 am; edited 1 time in total
Fri Jul 15, 2016 7:33 am  by bathfilms

Yes, sorry, I just realised that the axiom "... that ALL gloves come in PAIRS (one left, one right)" did not come from the original question, but rather from another reply.

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