Imagine that scientists in the rain forest have discovered a new species of plant life. Where might they look for the Fibonacci sequence?

Suppose that you're shooting baskets with a friend. After a few practice shots, you decide that you want to keep score. The first basket either of you makes is worth one point. Just to make things interesting, you suggest that every time either of you makes another basket, you add your previous two scores to get a new total. To make the game even more appealing, you offer to start from zero, while your friend can start from one. What sequence of numbers would emerge after shooting eight baskets? What is the difference in points between you and your friend? What pattern has emerged from the point difference?

Explain that numbers missing from the Fibonacci sequence can be obtained by combining numbers in the sequence, assuming that you're allowed to use each number more than once. For example, how could the number 4 be obtained from the sequence? How about 11? 56? Think of a number not in the sequence and try to figure out what numbers to combine to get it.

At first glance, the natural world may appear to be a random mixture of shapes and numbers. On closer inspection, however, we can spot repeating patterns like the Fibonacci numbers. Are humans more apt to perceive some patterns than others? What makes certain patterns more appealing than others?

Try to solve this problem: Female honeybees have two parents, a male and a female, but male honeybees have just one parent, a female. Can you draw a family tree for a male and a female honeybee? What pattern emerges? Are they Fibonacci numbers? (The male bee has 1 parent, and the female bee has 2 parents. The male bee has 2 grandparents, and the female bee has 3 grandparents. The male bee has 3 great-grandparents, and the female bee has 5 great-grandparents. The male bee has 5 great-great-grandparents, and the female bee has 8 great-great-grandparents. The male bee has 8 great-great-great-grandparents, and the female bee has 13 great-great-great-grandparents.)