17.6.2. Mapping to Ordered Pairs

Try these first

1. 1.

   What is the midpoint of the points (2, 4) and (4, 6)?

   Hint

   Use the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).

2. 2.

   How many even/odd combinations exist when evaluating coordinates?

   Hint

   Consider parity for x and y coordinates.

3. 3.

   What principle is used to guarantee that among 5 points, at least two share the same category?

   • Pigeonhole Principle
   • Midpoint theorem
   • Integer mapping

   Hint

   Think of categorizing items.

4. 4.

   True or False: The midpoint of (1,3) and (3,5) is an integer.

   • True
   • False

   Hint

   Calculate it to verify.

5. 5.

   Given 5 different integers in the range of 1 to 10, show how you can always find two that sum up to 11.

   Hint

   Chart out the pairs and check how they fit.

6. 6.

   Constructively prove that with any selection of 6 integers from 1 to 12, at least two integers will share the same parity.

   Hint

   Think about how many students can fit into the same two chairs.