The Math Kids: A Sequence of Events Page 2

With that threat, Justin had to give in. “What if zero factorial isn’t equal to zero?” he asked slyly.

  “How could it be anything but zero?” Stephanie asked.

  “My dad explained it to me,” Justin said. “He said factorials are used to figure out how many ways you can arrange the things in a group. For example, there are three of us. How many ways can we stand in a line?”

  I wrote the combinations down on a sheet of paper.

  Jordan, Justin, Stephanie

  Jordan, Stephanie, Justin

  Justin, Jordan, Stephanie

  Justin, Stephanie, Jordan

  Stephanie, Justin, Jordan

  Stephanie, Jordan, Justin

  “There are six ways to do it,” I said.

  “That’s right! Three people and six ways to arrange them, so three factorial equals six. Three times two times one,” Justin said.

  “That’s really cool,” Stephanie said.

  “I agree,” I said, “but it doesn’t explain why zero factorial is one.”

  “How many ways can you arrange zero people in a line?” Justin asked.

  “That doesn’t make any sense,” Stephanie said. “How can you arrange zero people?”

  “I guess there’s only one way,” I said. “One empty line.”

  “Exactly!” Justin said. “So, zero factorial equals one.”

  “I don’t know,” Stephanie said. “Still seems kind of strange to me.”

  “Well, strange or not, it solves the last part of The Sixes Problem,” Justin said.

  “How?” I asked.

  “Simple. If zero factorial is just one—” Justin started.

  “Then we can use zero factorial three times to get one plus one plus one,” Stephanie finished.

  “And that leaves us with three, and we know that three factorial is—” Justin began.

  “Six!” I shouted.

  I wrote the equation on the board:

  (0! + 0! + 0!)! = 3! = 6

  “Ha! Take that, Mrs. Gouche. The Math Kids strike again!”

  We had done it. We’d solved all the numbers in The Sixes Problem!

  Justin didn’t look pleased, though. Stephanie noticed and asked him why he wasn’t happy to solve the problem.

  “Oh, I’m happy we solved it,” he said. “It’s just that we didn’t do it on our own. We needed help from someone else. And not just anyone, either. Someone from the red math group.”

  We thought about that for a moment. It didn’t really matter to me that we had some outside help, but I guess I could see Justin’s point. We had cracked the Prime-Time burglar case all by ourselves, but this time we had needed help from someone outside of our club.

  “I’ve got a solution for that,” Stephanie said.

  Justin and I looked at her with raised eyebrows.

  “Let’s ask her to join the club,” Stephanie said.

  “Are you kidding me?” Justin asked. “She’s in the red group!”

  “Yeah, that’s weird, isn’t it?” Stephanie asked. “How can she be in the red group and still know cool stuff like factorials? It doesn’t make any sense.”oHo

  “Well, don’t ask me. I don’t know anything about her,” I said.

  “Then I’m going to make it my job to find out,” Stephanie said.

  CHAPTER 3

  Monday rolled around again. Mrs. Gouche was surprised to see that we had finished The Sixes Problem.

  “Nice job on this,” she said. “You even got the tough ones; zero, one, and eight are usually the ones that stump people.”

  I was happy to accept her praise, but Justin was still a little grumpy about not solving the problem on our own.

  As for Stephanie, she was on a mission to find out everything she could about Catherine Duchesne. Stephanie usually sat with us at lunchtime, but today I watched her go over to the table in the corner where Catherine sat all by herself. She was wearing a Josh Bell baseball jersey with a large 55 on the back. She was sketching a picture on a piece of notebook paper.

  “Hey, nice picture,” Stephanie said.

  Catherine looked up in surprise and quickly slid the paper back into her notebook.

  “Thanks,” she responded softly.

  “Can I sit with you?” Stephanie asked.

  “I guess,” Catherine answered.

  Stephanie sat across from Catherine. Neither said anything for a minute. Stephanie opened her lunch box to see what her mom had packed. Catherine picked at her food, not looking up at Stephanie.

  “Thanks for telling us about factorials,” Stephanie said.

  Catherine just nodded.

  “We couldn’t have done it without you,” Stephanie continued.

  Catherine nodded again.

  “I’m Stephanie.”

  “I’m Catherine, with a C.”

  “Where did you learn about factorials?” Stephanie asked.

  “I guess I read it in a math book.”

  “Well, I’m glad you did. We were really stuck.”

  “I saw it right away,” Catherine said. “I wish we got problems like that in our math group.”

  “I was going to ask you about that,” Stephanie said. “How come you’re in the red group?”

  Catherine shrugged her shoulders but didn’t answer.

  The rest of the lunch went pretty much the same way. Stephanie would ask a question and Catherine would say little or nothing. Sometimes she just nodded or shook her head. When the bell rang for the start of recess, Stephanie hadn’t learned anything at all about the mysterious Catherine Duchesne.

  As it turned out, we soon had other things on our minds. During math group that afternoon, Mrs. Gouche dropped a bombshell on the three of us.

  “The district is holding a Math Olympics competition this spring,” she announced. “Teams from each of the fourth-grade rooms will be competing to see who represents McNair.”

  “When is the competition?” I asked excitedly.

  “The fourth-grade rooms will compete in four weeks, so you’ll want to get started practicing right away,” she answered.

  “What’s the first step?” I asked. “What kind of problems will we be solving? How much time do we have? Will we—”

  “Hold on a minute, Jordan,” Mrs. Gouche said. “First, you need to find a fourth member for your team. Each team has to have two boys and two girls.”

  Justin and I looked dismayed. We already had a team. We didn’t need anybody else.

  Stephanie had another thought, and she blurted it out.

  “Catherine Duchesne,” she said.

  Mrs. Gouche looked surprised. Justin and I did, too.

  “I was thinking about Susie McDonald,” said Mrs. Gouche.

  No! Justin and I didn’t say it out loud, but if we had, we would have shouted it.

  “What about Catherine?” Stephanie asked.

  “These are really tough problems,” Mrs. Gouche said. “You know she’s in the red math group, don’t you?”

  Of course we knew that. We had just had a big discussion about her on Saturday.

  “Yes, but I’ve got a good feeling about her,” Stephanie persisted.

  “Well, we don’t have to turn in the team’s names until the end of the week, so think about it and let me know on Thursday. My recommendation is to go with Susie.”

  The competition, and the subject of the fourth person on the team, was all we talked about as we walked home that afternoon.

  “We can’t have Susie on the team,” Justin said.

  “You know my choice would be Catherine, but what do you have against Susie?” Stephanie asked.

  Stephanie was kind of new to our school, so I explained it to her. Susie’s mom is the president of the Parent Teacher Organization and I’ve heard my parents say that Mrs. McDonald thinks she rules the school. Because of her, Susie and her little brother Adam always get whatever they want. Susie got the lead solo in the school pageant when she was only in the third grade. Adam is the only second grader who starts every game on his socce
r team, even though there are lots of kids who are better players. I wondered if Mrs. Gouche recommended Susie for our team because she is afraid of what Mrs. McDonald might do if she didn’t.

  “I’m not sure I could convince Catherine to join the team anyway,” Stephanie said, “so it looks like we might be stuck with Susie.”

  Justin nodded sadly. “If Susie wants to be on the team, we’re definitely stuck with her,” he said. “Mrs. McDonald will see to that.”

  “Unless…” I said.

  “Unless what?” Justin asked.

  “Unless we can come up with a plan,” I said.

  “What did you have in mind?” Stephanie asked.

  I didn’t have a clue. I just knew we only had four days to convince Susie she didn’t want to be on the team, and to convince Catherine that she did.

  CHAPTER 4

  On Tuesday morning, Mrs. Gouche had a couple surprises for us. First, she moved our math group from the afternoon to the morning. That was fine with me. I’d do math all day long if I could. It sure beat English and social studies—not exactly my best subjects. The second surprise, however, was not fine. Mrs. Gouche moved Susie into the yellow math group. I could tell from the look on Justin’s face that he didn’t like it one little bit. I didn’t either.

  “I thought if Susie was going to be on your math team, then we should move her into your group so she can practice with you,” said Mrs. Gouche.

  So that was the way she was going to play it! She wasn’t really giving us a choice on who we wanted on our team. I bet that Mrs. McDonald had already talked to our teacher about it. Now we were really stuck!

  I glanced over at Justin, saw him make a face, and knew he had been thinking the same thing.

  “Here’s a practice problem for you guys to work on today,” Mrs. Gouche said. “There are a thousand people at a concert. Can you prove that there are at least two people who have the same first and last initials?”

  “That’s easy,” said Susie.

  We all looked at her in surprise. Maybe we had misjudged her math knowledge after all.

  “Can you prove it?” Justin asked.

  “Sure, it’s easy. All you have to do is go to each person at the concert and get their initials,” Susie answered, “and then compare all of them together to see if any of them match.”

  We just stared at her. Susie seemed quite proud of her solution. I didn’t have the heart to tell her how ridiculous it really was. Stephanie didn’t have a problem telling her how she felt, though.

  “That’s silly,” she said.

  Susie looked at her in surprise.

  “How else are you going to know for sure if you don’t ask them?”

  “It’s a math problem, so you have to use math,” said Justin.

  “Well, I’m sticking with my idea until you come up with something better,” Susie said in a huff. She refused to work with us on coming up with a solution to the problem using math.

  “Do you think factorials would work here?” I asked. “Isn’t this kind of like finding out how many combinations there are?”

  “Maybe,” said Stephanie doubtfully, “but I think there’s something more here.”

  Justin was looking at the problem. He had written it down as Mrs. Gouche described it. “She mentioned a thousand people,” he said, “so that number must be important.”

  Justin is great at pulling out the important pieces of information from a problem. Lots of times there are other things in the problem that don’t help you solve it, so you need to figure out which pieces are important and which ones aren’t. Justin is a whiz at that!

  He wrote 1,000 on the whiteboard. He also wrote 26.

  “What’s the 26 for?” Susie asked.

  Justin gave her a long look before responding. “There are twenty-six letters in the alphabet.”

  “So?” she asked.

  “So, there are twenty-six possible first initials and twenty-six possible last initials,” he said.

  “That would make twenty-six times twenty-six combinations of first and last initials,” I said.

  Stephanie did the math on the whiteboard. “That’s six hundred seventy-six possible combinations for first and last initials.”

  “And since there are a thousand people, at least two of them must have the same initials,” I said.

  “But how do you know that without asking them?” Susie asked.

  “Since one thousand is bigger than six hundred seventy-six, there must be at least two people who share the same initials,” I explained.

  “I don’t get it,” Susie said.

  “Isn’t it obvious?” Justin asked.

  “Not to me, it isn’t.” Susie glared at Justin.

  I tried to explain it to her. “There are six hundred seventy-six combinations of initials. A. A., A. B., A. C., and so on, then B. A., B. B., B. C., and so on, all the way up to Z. Z. Make sense so far?”

  Susie nodded, but she didn’t look very confident.

  “So, let’s say the first six hundred seventy-six people each have a unique combination of initials, from A. A. to Z. Z.”

  Justin and Stephanie were nodding along with my explanation, but Susie looked confused.

  I don’t think she was following my explanation, but I finished anyway. “When the next person comes up, they must have the same initials as someone else because we’ve already used up all of the possible combinations.”

  “So even if there were only six hundred seventy-seven people, at least two must have the same initials,” Stephanie said.

  “I still think it would be easier to ask everyone,” Susie said with a pout.

  “But that’s not proving it!” Justin protested.

  I could tell Susie and Justin were getting ready to argue, but Mrs. Gouche came over just in time to head it off.

  “How’s it coming?” she asked.

  “We got it solved,” said Stephanie, and went through the explanation for Mrs. Gouche.

  “Very nice,” Mrs. Gouche said. “That problem is an example of something called—”

  The lunch bell rang, cutting her off. The students scrambled for the door, and Mrs. Gouche stopped in the middle of her sentence to restore order in the classroom.

  At lunch, Stephanie sat down with Catherine Duchesne, who was once again sitting by herself at a table in the far corner of the cafeteria. This time she was wearing a Jarrett Jack basketball jersey. Like the baseball jersey she had been wearing the day before, this one also had a big 55 on the back. This time, Catherine was the first to speak up.

  “You guys were doing a pigeonhole problem in math group, weren’t you?” she said.

  “What kind of problem did you call it?” Stephanie asked.

  “Pigeonhole,” Catherine responded. “It’s a kind of problem where you can prove something by showing there are more pigeons than there are holes to put them in. If there are seven pigeons and only six pigeon holes, you can prove that at least one hole must have at least two pigeons in it. It’s like the sock problem.”

  “What’s the sock problem?”

  “You’re in a dark room. You open a drawer to get a pair of socks. You know there are ten red socks and ten blue socks. How many socks do you have to pull out before you know you have a pair?”

  “Eleven,” Stephanie blurted out.

  Catherine smiled. “Lots of people would say that, but it’s really only three. If you have three socks, you must have either three of one color, or two of one color and one of the other.”

  “I get it. Either way, you will have a pair,” Stephanie said.

  Catherine smiled and nodded. “Now, if you wanted to guarantee you have a pair of blue socks, you would need to pull out twelve socks, because the first ten might all be red.”

  Stephanie nodded thoughtfully. “Catherine, can I ask you a question?”

  “I guess,” Catherine said carefully.

  “Why aren’t you in the yellow math group?”

  The smile faded from Catherine’s face. She
didn’t say anything and instead went back to slowly eating her sandwich.

  “Hey, I didn’t mean to make you mad,” Stephanie said. “It’s just that you’re so good at math, I’m a little surprised.”

  Catherine slowly took another bite. When she finished chewing, she finally answered but did it without looking up at Stephanie.

  “Girls aren’t good at math,” she said quietly.

  “What?” Stephanie said in surprise. “Who told you that?”

  The recess bell rang. If Catherine answered, it was impossible to hear her quiet voice over the buzzing hive of activity, as students dropped their trash in the bins and headed toward the playground. Stephanie looked at the crowd pushing out the door, and, when she looked back, Catherine was gone.

  CHAPTER 5

  Stephanie didn’t get another chance to talk to Catherine the rest of the day. Justin, Stephanie, and I walked home together.

  “We’re down to two days,” Justin said with concern.

  “We need to get rid of Susie and add Catherine,” Stephanie said. “She said girls aren’t good at math, but she’s a whiz. She knew all about the problem we were solving this morning. She called it a pigeonhole problem.”

  “Why is it called a pigeonhole problem?” I asked.

  “Later. What’s important right now is getting her to join the team,” said Stephanie.

  “But if we can’t talk her into joining, what can we do?” I asked.

  “I still have two days to work on her,” Stephanie said with determination. “I’m going to convince her to join us. Your job is to find a way to get Susie to drop out.”

  She cut through the yard across the street from my house and was gone. Justin and I sat on my front porch and tried to figure out a plan.

  “If we can’t get Catherine, maybe we can just have Susie keep quiet during the tournament,” I said.

  “When have you ever seen Susie with her mouth shut?” Justin asked.

  I laughed, but Justin had a good point. Susie was always talking—before school, after school, and even during school. Anyone else would get in trouble for that, but Susie never did. I was sure Mrs. McDonald made sure of that. Since we couldn’t get Susie replaced, we needed a way to get her to leave on her own.