13.8. Question 7

Try these first

1. 1.

   What is the average of the numbers 1 to 10?

   Hint

   Sum the numbers and divide by the count.

2. 2.

   List any three consecutive integers from 1 to 10.

   Hint

   Select any number and follow with the next two.

3. 3.

   What is the average sum of three consecutive integers from any arrangement of 1 to 10?

   • 15.5
   • 16.5
   • 17.5

   Hint

   Think about how many times each number contributes to the sums.

4. 4.

   True or False: There exists a configuration of numbers from 1 to 10 where no groups of three sum to 17.

   • True
   • False

   Hint

   Utilize the average condition to analyze this.

5. 5.

   Using number representations 1-10, show how many distinct arrangements lead to various sums and highlight how many achieve above 17.

   Hint

   Think of iterative approaches to different rotational placements.

6. 6.

   What if you stretch the range from 1 to 15? How does that affect the applicability of our original theorem?

   Hint

   Expand on how three picks change the sums while bringing in higher figures.