### Brain Teasers

# Mr. Froopaloop's Socks

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

Hans Ernest Froopaloop, Jr. always buys or receives socks in pairs, and each sock in each pair is always the same colour, which is either red or green. Lucky Mr. Froopaloop has only ever lost one sock in the laundry.

After that sock was lost, the remaining socks gained a mathematical property. If Mr. Froopaloop's socks are all unpaired and all in the same drawer, and if he takes two random socks out of the drawer, the probability of the two socks being the same colour is exactly 50%.

If there are more than 250 but less than 350 socks in total, and if there are more red socks than green socks, what colour sock was lost?

After that sock was lost, the remaining socks gained a mathematical property. If Mr. Froopaloop's socks are all unpaired and all in the same drawer, and if he takes two random socks out of the drawer, the probability of the two socks being the same colour is exactly 50%.

If there are more than 250 but less than 350 socks in total, and if there are more red socks than green socks, what colour sock was lost?

### Hint

There has to be an odd number of socks.### Answer

Mr. Froopaloop lost a red sock.Let x be the number of ways Mr. Froopaloop can take out two socks of the same colour. Let y be the number of ways Mr. Froopaloop can take out two different-coloured socks. Since the probability of the two socks being the same colour is 50%, then:

x=y

Let r be the number of red socks, and g be the number of green socks. If the first sock Mr. Froopaloop takes out of his drawer is red, he will then have r-1 red socks in the drawer, but still g green socks. So he has r*(r-1), or (r^2)-r, ways of taking out two red socks, and r*g ways of taking out a red sock first and a green sock second. Using similar logic, we can determine that he has g*(g-1), or (g^2)-g, ways of taking out two green socks, and g*r, or r*g, ways of taking out a green sock first and a red sock second. Therefore:

x=(r^2)-r+(g^2)-g

And:

y=(r*g)+(r*g)

y=2*r*g

Combining this with our first equation:

(r^2)-r+(g^2)-g=2*r*g

(r^2)-(2*r*g)+(g^2)=r+g

(r-g)^2=r+g

This means that the total number of socks equals the square of the difference between red socks and green socks. Since the socks are always bought or received in pairs, there must have originally been an even number of socks. But since only one was lost, there must be an odd number now. The only odd square number between 250 and 350 is 289, so this must be the total number of socks, and there must be sqrt(289), or 17, more red socks than green ones. Therefore:

r+g=289

r-g=17

Solving for these equations, we get r=153 and g=136. Since there is now an odd number of red socks and an even number of green socks, the lost sock must be red.

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