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### Multiplying Mice

```
Date: 07/23/97 at 23:02:08
From: Anonymous
Subject: Multiplying Mice

Marco's Uncle Oscar gave him two mice, which he named Whiskers and
Oscar. But Marco discovered that he made a big mistake. Oscar should
have been named Oscarella - she just had 8 babies, 4 males and 4
females! "Ten mice aren't so many," said Marco to his mother. "They're
cute."

"Yes," said his mother, "but these cute baby mice can breed when they
are 6 weeks old. The babies are born after 3 weeks. If each mother
mouse has only one litter and all the litters have 8 babies, half
males and half females, how many mice will you have 18 weeks from
today?"

What answer should Marco give to his mother?
```

```
Date: 07/24/97 at 13:16:55
From: Doctor Rob
Subject: Re: Multiplying Mice

This is a nice twist on the old problem of Fibonacci about breeding
rabbits.

The number of mice at any time in the future is the same as the number
of mice three weeks earlier (survivors) plus four times the number of
mice nine weeks earlier (breeders).  This leads to the following
recursion:

x[n] = x[n-3] + 4*x[n-9].

The initial conditions are that x[-6] = 2, x[-3] = 2, and x[0] = 10.
You want x[18].  Apply the recursion six times, with n = 3, 6, 9, 12,
15, and 18.

Hint:  The answer has three digits.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 07/24/97 at 16:13:51
From: Anonymous
Subject: Re: Multiplying Mice

Thank you!
```

```
Date: 07/25/97 at 15:34:46
From: Doctor Rob
Subject: Re: Multiplying Mice

I'm sorry that you didn't understand the answer I gave.  I didn't just
give the answer because we show people how to solve a problem without
doing it for them. You will learn more from doing it yourself with
some direction from us than if we just tell you the answer.

Make a list of times starting with 6 weeks ago.  Next to it, make a
list of the number of mice that existed at that time.

Time           Number of Mice
====           ==============
6 weeks ago          2
3 weeks ago          2
Today               10
3 weeks from now    18 = 10 + 4*2
6 weeks from now    26 = 18 + 4*2
9 weeks from now    66 = 26 + 4*10

The number of mice in any line in the above table is the number in the
line above it (mice that survive from one line to the next), plus the
number of new babies.

The number of new babies is four times the number of parents who are
producing offspring. Since it takes parents 9 weeks from birth to
produce new offspring, the number of fertile parents is the number of
mice 9 weeks earlier, which appears three lines above.

To calculate the number in the next line, add 66 (from the line above)
to 4 times 18 (from the 3rd line above). Repeat this operation until
you have the answer for 18 weeks from now.

Is that better?

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 07/25/97 at 23:04:34
From: Anonymous
Subject: Re: Multiplying Mice

Yes, but I still don't understand about breeding and litters. Can
do the babies wait to make more babies?

I came up with a answer of 506, but it's probobly wrong. Can you
explain the problem and how to figure it out?

Thanks!
```

```
Date: 07/28/97 at 09:25:12
From: Doctor Rob
Subject: Re: Multiplying Mice

I think we have isolated the problem. You are not familiar with the
terms used in stating the problem.

For a pair of mice to breed is to begin the process of having babies.
Another word that is used is that the mice mate with each other.

A litter of baby mice is the set of babies that are born together at
the same time to one mother.  The same word is used for puppies and
kittens.

Yes, parents who have had children can have children again in 3 weeks.

Babies have to be 6 weeks old to breed, and 9 weeks old before their
babies can be born.

The idea is figure out how many new babies are born at each 3-week
interval. Then you add them to the number of mice that existed 3 weeks
earlier to get the new number of mice. The number born is 8 for each
pair of parents, or 8/2 = 4 times the number of parents. The number of
parents is the number of mice born at least 9 weeks ago, since it
takes 9 weeks from birth before mice can give birth to their own
babies.

Putting this together, the number of new babies is 4 times the number
of mice born at least 9 weeks earlier. Add this to the number of mice
3 weeks earlier and you will get the new number of mice.

This suggests building the table above, with each line representing
the situation at a particular time. Since all the activity takes place
in multiples of three weeks, the first column of times in the table
are spaced three weeks apart. Then the first three rows in the second
column can be filled in from the statement of the problem, with
Whiskers and Oscarella and their first litter of babies (2, 2, and
2 + 8 = 10).

For the rest of the rows in column 2, use the rule figured out in the
previous paragraph to calculate the number.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Fibonacci Sequence/Golden Ratio
High School Sequences, Series
Middle School Word Problems

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