# Pairing problem

• Aug 17th 2012, 10:14 PM
mjoshua
Pairing problem
Each of the 75 children in a line was assigned one of the integers from 1 through 75 by counting off in order. Then, standing in the same order, the children counted off in the opposite direction, so that the child who was assigned the number 75 the first time was assigned the number 1 the second time. Which of the following is a pair of numbers assigned to the same child?

How can I show and prove that it has to be 47 and 29?
• Aug 17th 2012, 10:31 PM
GJA
Re: Pairing problem
Hi, mjoshua.

It might help to think about what happens when you add the numbers together that a student received.

Does this help get things on the right track?

Good luck!
• Aug 17th 2012, 10:35 PM
mjoshua
Re: Pairing problem
It does not, I'm sorry.. Anyone else?
• Aug 17th 2012, 10:40 PM
GJA
Re: Pairing problem
Just for fun, can you write the two numbers assigned to person number one, the two numbers assigned to person number two, then add each pair of numbers...I think that might move things in the right direction
• Aug 18th 2012, 09:42 AM
mjoshua
Re: Pairing problem
So the proof is the pairs must add to 76?? That seems rather bland..?
• Aug 18th 2012, 10:43 AM
Soroban
Re: Pairing problem
Hello, mjoshua!

Quote:

Each of the 75 children in a line was assigned one of the integers from 1 through 75 by counting off in order.
Then, standing in the same order, the children counted off in the opposite direction,
so that the child who was assigned the number 75 the first time was assigned the number 1 the second time.
Which of the following is a pair of numbers assigned to the same child?

How can I show and prove that it has to be 47 and 29?

Take a look at GJA's observation . . .

. . $\begin{array}{c|cccccc} \text{1st count} & 1 & 2 & 3 & \cdots & 74 &75 \\ \hline \text{2nd count} & 75 & 74 & 73 & \cdots & 2 & 1 \end{array}$

Each child received a pair of numbers whose sum is 76.

Yes, that's kind of bland.
But when an answer is that simple, don't expect fireworks . . .