Mathematics is the study of patterns and problem solving. As I went further in my graduate studies, I became more and more intrigued by the patterns aspect of mathematics.

**What’s the next number in the sequence. What’s the 5th number in the sequence?**

These are intriguing questions to solve for any pattern. But the ultimate question to solve is what is the pattern for the millionth or billionth number in the sequence? To answer that question, you have to crack the code or the **equation** that gives you the answer for any term in the sequence. We call this “any term” the **nth term**.

For example, what’s the next number in this sequence?

5, 7, 9, 11, 13, 15

If you figured it was 17, you are correct.

But can you tell me what the 25th number in the sequence is? Or the 100th number?

Using number theory, you can determine that the 25th term is 53:

If the 5th term is 13 and the 6th term is 15, then the code is twice the term plus 3. So twice 25 plus 3 is 53.

Expressed in terms of any number, 2n+3.

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**So how would you unravel the pattern for intersecting circles? **

My thoughts are to start with a smaller problem. Figure out the number of possibilities for three circles, then for four circles. Use manipulatives to more easily visualize the possibilities: coins, bingo chips, circles cut from different colored post-its. Record your data in a table for n=2,3 or 4 and crack the code.

**Please share your thinking for how you solved this—I am more interested in your thinking and your problem solving approaches then in just the answer.**