Extreme Math
Home Study Assignment #4
Here’s a problem that has been challenging my top tenth-grade students:
| Of the numbers in the 100th row of Pascal’s Triangle, how many are odd? |
You’re not going to solve this by carefully writing 100 rows of Pascal’s Triangle! After all...
- It would take a ridiculous amount of time.
- You would make mistakes.
- You wouldn’t learn much from doing it that way anyway.
You’ll learn something (and save lots of time) by looking for patterns in the information you already have. Here are some questions that will lead you to the answer to the problem in the box:
- Make a chart showing how many odd numbers there are in rows 0-16, using a copy of Pascal’s Triangle that you already have. If it helps, you may want to use the copy where you have already colored in all the odd numbers. Write your results here:
| row #: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
| How many odd numbers? |
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- Do you see any pattern yet? If not, focus on the Mersenne numbers (rows #1, 3, 7, 15, 31, etc.).
- Now continue Pascal’s Triangle to row 25 or so, just writing O for odd numbers and E for even. In each place, think about what an odd number plus an even number is, or what two odd numbers add up to, or what two even numbers add up to.
- Now see if you have enough information to find the pattern. As a hint, draw a thick line right before every row number that’s a power of 2. If you can’t yet find the pattern, continue up to row 31 and try again.