17.4. Tutorial 6: Part II

Try these first

1. 1.

   What is the midpoint of points (2, 4) and (4, 8)?

   Hint

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

2. 2.

   Choose any three integers from 1 to 8 and see if you can find at least one pair whose sum equals 9.

   Hint

   Look for combinations that add up to 9.

3. 3.

   What does the pigeonhole principle state?

   • More items than containers lead to duplicates
   • All items can fit uniquely
   • Items can be distributed evenly

   Hint

   Think about objects in boxes.

4. 4.

   True or False: If two points have even coordinates, their midpoint will also have even coordinates.

   • True
   • False

   Hint

   Calculate to check the conclusion.

5. 5.

   Using the pigeonhole principle, prove that if there are 10 students in a class with 4 different hair colors, there is at least one color shared by at least three students.

   Hint

   Think about distributing more items across fewer categories.

6. 6.

   Show that among any ten integers selected from 1 to 20, there must be a pair whose sum is 21.

   Hint

   Count and see which pairs complete the set.