This question was first asked nearly 800 years ago.
“In a land far away, King Fudge and Queen Ramona, guided by the wisdom of Red Bunny, peacefully ruled their people.” So began the first session of our spring Math Circle for young children. With the help of my assistant R, I told the story of this land. The first chapter culminates with a contest for sorcerers to match wits against Red Bunny, who is retiring. The winner will take over Red Bunny’s job. The king and queen aren’t sure how to design a test to choose an advisor, so they seek advice from Red Bunny and his friend Orange Bunny. “So they need advice about advice!” noticed A.
“What’s the test?” pleaded P several times throughout the story. After setting up the scenario in detail, we finally got to the actual test: “A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in 6 months if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?”* R told the kids that the sorcerer’s answer was “twelve.”
“Is he right?” I asked. No one was sure what the question meant. M took a stab at clarifying: “They put 2 rabbits in a place, and they are asking him how many babies there are.” Understanding dawned. One person then answered yes, the sorcerer is right, since the answer is 2 rabbits times 6 months, and 2 times 6 is twelve. Someone else contradicted: no, it’s 12 because it’s 6 plus 6.
“They’re the same thing!” exclaimed V.
“I don’t get it,” said L, “couldn’t there be two female rabbits at the same time?”
P added another objection: “Normally, bunnies can have 500 kids at a time.” We spent some time clarifying the problem, coming to terms with these and other illogicalites. I asked again whether the sorcerer gave a correct answer, and, while most kids still thought yes, a bit of doubt emerged. So we worked through the problem with multiple colors of chalk on the board. This took a loooooooooong time. The conversation went like this:
“So, where did this group of bunnies come from?” (me, pointing to some roughly scribbled bunny heads)
“The pink pair.” (someone)
“I don’t think so. What if this new group was born at the end of the month?” (V)
The problem turned out to be way more complicated than it initially appeared. I told them that this question was first asked (in our world) nearly 800 years ago and people are still enjoying it. The group discussed the complications, agreed on the numbers, and figured out the third and fourth months without too much toil. Their ease in this part of the calculation gave them the confidence to form some conjectures about the final (six-month) number. “There will be 13 pairs,” said J. She started to justify her number but was contradicted mid-explanation by a few other kids. D kindly rescued her with his comment, “There could be 13 pairs, I guess.” We decided to see what happens in the fifth month before accepting other conjectures about the sixth month. (Earlier in the class, V, who is new to math circles, asked what a conjecture is. M and L, math circle veterans, explained.)
“In the fifth month,” explained E, who had come up to the board and used her pencil as a pointer, “the white ones will have babies. I don’t think the pink ones will, because they already had 2 pairs.” Most of the others agreed.
“We are spending too much time on this silly problem,” exclaimed P. “We already know it’s 12!” This kind of frustration comes up sometimes when kids are pulled in one direction by the drama of the narrative and in another by the desire to solve the problem. P wanted to know what happened with the sorcerer, and what the next test will be. His interest came back toward the math when I asked the whole group, “Do we know?” and many voices answered loudly “Nooooo!”
We went rabbit pair by rabbit pair asking “Can the white (purple/pink/blue/green….) rabbits have babies in this month?” Their answers led most of the kids to believe that there would be 8 pairs of rabbits by the end of the sixth month. There was some confusion, though, as to whether each month’s total should be added together for the final answer. It seemed that almost every child in the room was either asking or explaining. E, P, D, and J came up to the board to explain. At one point in time, it seemed that all 9 members of our collaborative agreed that the answer was 8 pairs, and that the sorcerer was wrong.
“So,” I inquired, “Should this sorcerer advance to the next level?”
“NO!” shouted the kids. They wondered what the next sorcerer’s challenge would be, and I told them that it would be this same challenge, but with more months, no more room on the board to draw bunnies, and no more colors of chalk.
“I’ll just get my mom to look it up on her phone,” said P. Someone else had suggested this about a non-mathematical question earlier in the class.
“Would the phone, or google, understand the question? Think about how you would phrase that,” I requested. We wondered together whether a piece of technology could understand such a convoluted question.
We were out of time now, so some kids joined their parents in the hall. Some stayed to stare at the board and think about how the numbers would continue. L came up to the board and asked me to explain the answer we had gotten. It turned out that in month six, I had run out of different colors of chalk, and used the yellow for the offspring of three different pairs, violating the color convention I had used for the first five steps of the problem. This change did alter the answer if the colors were meant to be consistent. L caught this; I hadn’t. J realized what I had done, and started to make color corrections on the board. Finally satisfied with the solution, L left the room. V remained, and figured out the seventh month in her head.
I’m looking forward to continuing the adventure next week. Please email me directly if you’d like for me to send you the story in detail.
*This question is a translation of the original, by Fibonacci. The phrase “6 months” was “a year” in the original.