Road Race Brain Teaser

A road race has a total of x people entered. After paying all his expenses, the race director's profit is x dollars per entry. He decides to use this money to buy T-shirts for the two running clubs that volunteered to help with the race. With the profits from the race he buys all the $12 T-shirts he can afford, and with what remains(less than $12) he buys a pair of socks. After dividing the T-shirts between the two clubs he finds that the first club has one more shirt than the second club, so he gives the second club the pair of socks. However, he realizes that the socks are not worth as much as a T-shirt, so with his own money he buys a pair of running gloves and gives them to the second club to make up the difference. How much are the gloves worth?

--- This puzzle is excerpted from Mathematical Morsels by Ross Honsberger who in turn got it from American Mathematical Monthly, 1930, p. 162, Problem 3379 proposed by J.H. Neeley and T.L. Smith, Carnegie Institute of Technology. However, I changed the original story to a road race story.

$1

about tree fiddy

$8

please post the reasoning for an answer. with the difference only being told as "less than $12" when he buys the sox then how can you know. say the difference was 8 say it was 5, say it was 1. then it's still different so he buys a pair of gloves. i don't see how you can come up with a definate, i'd like to see the explanation so that i can feel like a total dumbass.

No patience to try to solve this. What I want to know is why he's overpaying for the t-shirts! If you can't find printed t-shirts for less than $6.50 you're not trying.

This is how I look at it, which is probably wrong.

For example there are 200 runners.

The RD makes $2 per runner.

He has $400.

He buys 17 shirts for club A $204

He buys 16 shirts for club B $192

This comes out to $396

This is where it gets confusing....

If the socks are $1 then he has to pay $11 for gloves

If the socks are $2 then he has to pay $10 for gloves

What's the answer?

x entries at x profit per entry means the available money must be a square ie, 1,4, 9, 16, 25, 36, 49, 64, 81, 100, 121, etc. He is able to purchase an odd number of T shirts (can't be divided equally between 2 teams) Th cost of the shirts must be an odd multiple of 12 ie, 12, 36, 60, 84, 108, 132, 156, etc. The difference between the profit and the t shirt cost must be less than 12. The only numbers that produce this result are 16 (4 x 4), 64 (8 x 8, or 4 x 4 x 4), 256 (16 x 16 or 4 x 4 x 4 x 4), etc . In each case the remainder when the cost of the tshirts is subtracted is exactly $4, which means the gloves cost $8 (12 - 4).

No idea on the cost of the gloves, but the RD is priceless.

i think you're right. the exception being 6 entries, 3 t-shirts, a free pair of socks and $12 gloves.

I thought of that one, but assumed that there had to be at least $1 left as he purchased the socks...

Also, with 6 entries, 3 T shirts, etc, there is more than one solution which doesn't make for a nice mathematical solution. I can't be bothered to try to express it as a formula, but i'm sure it can be done:)

math whiz wrote:

The only numbers that produce this result are 16 (4 x 4), 64 (8 x 8, or 4 x 4 x 4), 256 (16 x 16 or 4 x 4 x 4 x 4), etc .

That's not true. You are saying that x=2^n for n>1 are the only values of x that satisfy the constraints. 20 and 892 also satisfy the constraints and are not given by 2^n (2 to the power n). To be sure the gloves cost $8, you have to show that all values of x that satisfy the constraints give a remainder of $4.

Phaedrus,

Saying he buys a pair of socks implies that the money left after buying T-shirts is not $0.

I'm bumping this back to the first page because I thought people on the west coast might want to give it a try.