Multiplying MiceDate: 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 Can you please simplify it the answer? 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 parents that have already made children make children again? How long 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. Your answer of 506 is correct. 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/ |
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