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Fibonacci Math Games

Fibonacci numbers are a math sequence named after Leonardo Fibonacci. He developed it while imagining the number of rabbits born in a year under certain conditions. The sequence is 1, 1, 2, 3, 5, 8 and so on. From the third term on, each number is the sum of the two proceeding numbers. The formula is F(n) = F(n-1) + F(n-2), for n >= 3.

Fibonacci numbers occur naturally throughout nature, such as in pineapple spirals or flower petals. They can be used as the basis of enjoyable math games.

Candy Machine
- A candy machine can accept a combination of quarters and half dollars. Calculate how many ways (n) the money can be arranged to in order to buy candy.
  
  This game can be played by using items such as play money or checkers to represent the coins. By forming piles and recording the results in a chart, it is easy to see that the patterns form a Fibonacci sequence. The chart should display the cost, number of multiples n, number of ways to pay f(n), and the patterns in exact order.
  
  If the candy costs 25 cents, then only one combination can be used (Q). At 50 cents, there are two: two quarters (QQ) or one half dollar (H). For 75 cents, there are three: three quarters (QQQ), a quarter and a half dollar (QH) or a half dollar and a quarter (HQ). For one dollar, there are four: four quarters (QQQQ); two quarters and a half dollar (QQH); a half dollar and two quarters (HQQ); a quarter, half dollar and a quarter (QHQ); or two half dollars (HH).
  
  The sequence is 1, 2, 3 and 5 for the numbers 1 to 4, and follows the Fibonacci pattern as more coins are added.
Flower Garden
- A bumblebee espies a garden with two rows of flowers and proceeds to visit each. He always begins at the left end, and can only travel in straight vertical or horizontal lines and never on a diagonal. He can only go forwards and never backwards. How many ways (n) can he travel if he visits one or more flowers?
  
  Draw two rows of dots. Label the top row 1 and the bottom row 2. For each dot, use a letter. Thus the first dot in row 1 is 1A, and the third dot in row 2 is 2C. Use a pencil to connect the dots as the bee travels. The chart should show the number of visited flowers visited (n), the exact order of patterns, and the number of ways f(n).
  
  If the bumblebee visits one flower, the number of ways he can travel is 1 and the pattern is 1A. If the bumblebee visits two flowers, he has two paths: 1A-1B, where two dots are connected to form a horizontal line, and 1A-2A, where two dots in the first and second rows are connected to form a vertical line.
  
  If the bumblebee visits three flowers, there are 3 paths: 1A-1B-1C, 1A-2A-2B, and 1A-2A-2B. The sequence is 1, 2 and 3 for the numbers 1 to 3, and follows the Fibonacci pattern as more flowers are visited.
Stacking Checkers
- An (n)-story stack of red and black checkers, labeled R and B respectively, is to be built in such a way that no two adjacent stories can be black, though they can be red. Find the number of possible ways a(n) that stacks can be created for (n) stories where n>=1. Pennies and dimes may be substituted for the checkers.
  
  For 1 story, two possible stacks are R and B. For 2 stories, there are three: RR, BR, and RB. For 3 stories, there are 5: RRR, BRR, RBR, RRB, and BRB. The sequence is 2, 3, and 5 for the numbers 1 to 3, and follows the Fibonacci pattern as more checkers are stacked.

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