# A Bee Family Puzzle Makes the Fibonacci Sequence Easy and FUN!

A beautiful honeybee named Lisa smiled at her adopted baby brother as he slept in his crib. After her father flew into the room, she asked him, “Why don’t boy bees have fathers?” He responded, “That’s just how nature made us. Male bees only have a mother while females have a mother and a father.” She asked, “Don’t you wish you had a father?” “No,” he replied. “Your grandmother was wonderful. She made me very happy. You know, this discussion gives me an idea for a cool, new puzzle. To solve the puzzle, you must draw a picture that shows how many grandparents your mother and I have over three generations.

“First, you need to draw a family tree for me and then one for your mother. You’ll draw an “M” for me and an “F” for your mother. Then draw a line to connect us to our parents (or parent for me). Next, draw a line from each parent to their parents (our grandparents). In each generation, you will use an “M” for grandfathers and an “F” for grandmothers. Okay, try it now. When you finish, I’m going to bring home a big bowl of honey for you and your new baby brother.” “Mmmm, yummy, I can’t wait!” Lisa said.

LISA’S BEE FAMILY TREE

(Child should draw one line from “M” to “F” and two lines from “F” to “M/F.” Connecting lines start from the top line.)

`Lisa's Parents:  M (Lisa's father)       1      F (Lisa's mother)      1Their Parents:   F (his mother)          1      MF (her parents)       2Gen 1 Grandp:    MF                      2      F MF                   3Gen 2 Grandp:    F MF                    3      MF F MF                5Gen 3 Grandp:    MF F MF                 5      F MF MF F MF           8`

Excellent job!! You drew two perfect bee family trees!! Now, how many grandparents are in the first generation? Yes! 5. What about the second? Perfect! 8. And the third? Correct! 13. Great job!! Now I’m going to show you a secret order of numbers that – like magic – automatically lets you know how many grandparents are in each generation.

The pattern for these numbers (called Fibonacci numbers) is: 1, 1, 2, 3, 5, 8, 13… Male bees start with the first “1” because they only have one parent, a female, and she’s the second “1.” For females, the pattern starts with the second “1” because they have two parents who represent the “2.” Here’s how Fibonacci numbers enable you to play a trick on your friends. Show them the drawing you just did. Then ask them to help you draw two more generations. After everyone draws two more generations, ask them to play the Bee Guessing Game.

Each person has to guess how many bee grandparents are in the fourth and fifth generations in two seconds per guess. You tell everyone that it’s against the rules to count each “M” and “F.” The winner gets to keep the family tree drawing. Undoubtedly, everyone – except you – will guess the wrong numbers.

You “guess” there are 21 grandparents in the fourth generation and 34 grandparents in the fifth generation. You get the right answers because you wrote the Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, 21, 34… ) on a small piece of paper that you taped to the side of your table. You write the sequence starting with the “3” because the first generation of grandparents for both of Lisa’s parents contains 5 members.

The Fibonacci Rule says each number is the sum of the two previous numbers. Therefore, the fourth generation totals 21 bees because 8 plus 13 equals 21. The fifth generation has 34 members, since 13 plus 21 equals 34. This Sequence works for every male bee and every female bee. Males start with “1” because they have only one parent. Females have two parents so they start with the second “1.”

For Lisa’s father’s third generation, you start counting to 3 from the second “1,” and get 5 grandparents. What about the third generation for Lisa’s mother? Correct! Starting from the “2,” you count to 3 and get 8. How cool is that!