Problem Solving Strategies: Finding Patterns

#1: (From Mathcounts handbook) Find more information on Mathcounts here -- the best middle school math competition program up to the national level.
A 16-page booklet is made from a stack of four sheets of paper that is folded in half and then joined along the common fold. The 16 pages are then numbered from front to back, starting with page 1. What are the other three page numbers on the same sheet of paper as page 5?\


Solution: It’s always a good idea to simplify a question when it’s totally new and see if you can find the pattern.
Fold a sheet of paper in half and mark the page numbers: 1-2-3-4
What have you got? Page 1 is always followed by page 2, page 3 by page 4….. When page numbers increase, it follows the same pattern, so page 77 will be followed by page 78, etc…
Another thing to notice is on the same side, page 1 is opposite of page 4 and page 2 is opposite of page 3. 1 + 4 = 5 The same goes to 2 + 3 Now we can solve this problem.

Since this is a 16-page booklet, page 1 will be opposite of page 16, so the sum of the numbers on the same side will all equal to 17.
Page 5 will be followed by page 6. The opposite number of page 5 would be 17 – 5 = 12 and the number opposite 6 would be 17- 6= 11
The answer is 6, 11, and 12.

#2: (Online brainteaser): The girls in gym class today sat around a large circle and spaced themselves evenly. The gym teacher asked them to count aloud in order, starting with 1, 2, 3, and so on. When they were finished counting, the girl who was 21st was sitting directly across from the girl who was 7th. How many girls were sitting around the circle?


Solution: Again simplify the question. Let’s make it 6 girls. In this case 1 will be opposite 4 and 2 will be opposite 5, 3 will be opposite 6.

How do we get 6 girls? Observe the number pairs and you will see if you multiply the difference by 2, you’ll get 6 each time. 2 (4-1) = 2(5-2) = 2(6-3)

So the answer to the question would be 2(21-7) = 28