## Pairing

Posted on: March 18th, 2014 by
1

You want to pair 2n students into n pairs of 2 students each. You like to do it on consecutive days so that no two students are paired together twice. What is the maximum number of consecutive days for which you can achieve such a pairing?

#### One Response to Pairing

1. Mike Earnest had this to say about that:

2n-1 days. To do this, arrange everyone except Sam in a regular polygon with 2n-1 sides. Number the kids 1,. . .,2n-1 consecutively. On day k, make two kids partners iff the line through them is parallel to the line through k and k+1; this pairs everyone except one person, who gets paired with Sam.

For example, if 2n=6, then 5 kids who aren't Sam are arranged like so:

4
5 3

1 2

On Day 1, the pairs are (1,2) and (5,3), leaving 4 with Sam.
On Day 2, we match (2,3) and (1,4), leaving 5 with Sam.
. . .
On Day 5, we match (5,1) and (4,2), leaving 3 with sam.

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