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Multiplying Rabbits 

Problem Statement

A newly born rabbit is capable of reproducing at one month old (when it matures). Suppose the rabbit never dies, and it continues reproducing one new rabbit every month. So, when the rabbit is born, it has one member in its own family. After a month, it matures, and by the second month it adds a new born member to its family. In the third month, the rabbit produces another offspring; its first child also matures and will be ready to have an offspring in the next month.  The sequence named by Fibonacci (1,1,2,3,5,8,13,21,...) can describe the number of members in the rabbit's family at each month. Explain how.

Problem setup

If a rabbit matures in one month and gives birth to at least one female rabbit the following month and every month thereafter, and every rabbit produced follows the same pattern, then what is the sequence that plays out?  The problem goes ahead and tells you that the sequence relates to Fibonacci’s sequence (In reality, it IS Fibonacci’s sequence).  It is a simple pattern like many others.


Plans to Solve/Investigate the Problem

First step:  Draw a diagram of how the process would materialize to a certain point (5 months or generations of rabbits) using color codes to differentiate between mature rabbits (signified by a blue letter “M”) and new rabbits (signified by a red letter “N”). 


Investigation/Exploration of the Problem

Having been informed in the problem that Fibonacci’s sequence was utilized in the problem, I could quickly see that the number of total rabbits from the last month quantified added to the total number to the previous month will give you the number of rabbits in the next month.  For example, in month 4 you will have five rabbits and in month five you will have 8 rabbits total and in month 6 you will have 13 rabbits.  We wanted to determine how many rabbits would be present in the 100th month but we didn’t want to work each month to find it out so we used Microsoft Excel.  We put in the values for the first two months and then put in a formula that employed Fibonacci’s sequence (example:  B4=B2+B3).  We then dragged the formula down the page to reach the 100th line or 100th month.  We discovered that at line number 56, which would be the sum of the rabbits in line 54 and the rabbits in line fifty-five, the answer was in scientific notation due to a number of digits higher than the computer would express (the answer was 1.39584E +11 or 1.39584 x 10 to the 11th power.) 


Extensions of the Problem

We decided that to get the ratio from month to month, we would divide each answer for the total number of rabbits by the answer from the following month (example:  In month two there is one rabbit and in month three there are two rabbits.  When dividing the earlier month by the following month we get 0.5).  We used Microsoft excel and applied the formula (B2/B3) from the beginning to the hundredth month and after learning that you do not have to manually drag the mouse (you can just place the cursor on the southeast corner of the frame of the last determined answer and to get the necessary symbol, a black plus sign, and then double click) to get the formula to run throughout the entire listing.  We were interested to find that at the 25th line (or the 25th month) you get a figure that begins repeating.  The number is 0.618033989, which is the golden ratio – something I had never heard of before.  We discussed the ratio and how it is in many aspects of life including the human body. 

Author & Contact
Kevin Smith
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Link(s) to resources, references, lesson plans, and/or other materials
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