by David Cole
4. In the addition problem below, each letter represents a digit. What four-digit number does DEER represent?
5. To write the numbers from 12 to 14, we will write a total of six digits (1 2 1 3 1 4). How many digits will it take to write the numbers from 1 to 150?
We had talked before the competition about how to work together in the team problem-solving round. There were a few ways we could go; We could each solve a different problem, and then whoever finished their problem first could start on the last problem. Or, we could work on each problem as a team. We had tried both these methods during our math group practice. What we had figured out is that we were best when we divided into two groups of two. That way, no one would get stuck on a problem by themselves, and we would have a second person to check to make sure the answer was correct.
“Catherine and I will take the first two,” said Stephanie.
“That leaves problems three and four for us,” I said to Justin.
“We’ll meet you at problem five,” Justin said. And the race was on!
As it turned out, we all finished at almost the same time, so we all got to work on the last problem together. It was a perfect team problem, so we finished it quickly. We looked at the clock. We had finished all five problems and still had five minutes to go. We saw Joe glance up from his paper and check the time nervously. The three other members of his team had their heads down working, but it was clear they weren’t working together as a team.
“Time!” shouted Coach Harder.
We waited while the judges checked the answers for each team. Mrs. Robinson’s class had three right, while the team from Mr. Kinney’s class only managed two. When they wrote up the score for Joe’s team, they wrote a big 40 on the board. His team had only gotten four right! If we had a perfect score on the team problem-solving, we’d get fifty points and we’d be tied!
The judges bent over the table, checking our answers while we waited anxiously. Coach Harder picked up the blue marker and stood in front of the board. He wrote our score and then stepped aside so everyone could see. A perfect fifty! We were tied with Joe’s team!
It was all going to come down to the lightning round. It was the Math Kids versus Joe’s team. Each team had eighty points, so no one had an advantage going into the round.
“Okay, teams, if you can take your place at your tables, we’ll start the final round of the competition,” said Coach Harder into his microphone. “During this round, each correct answer will be worth five points. After I’ve finished reading the question, ring your bell when you have an answer. If your answer is correct, you’ll be awarded five points. If not, the other team will be allowed to answer. The round will last for ten minutes. If the two teams are still tied at the end of the round, we will go into sudden death. Does everyone understand the rules?”
Both teams nodded.
“Are both teams ready to start?”
We nodded nervously. Joe, on the other hand, looked supremely confident.
“Start the clock. Here is the first question. I have forty socks in a drawer: twelve tan, nine brown, eleven gray, and eight blue. It is pitch black. How many socks do I need to pull out to guarantee I have a matching pair?”
Stephanie slammed her hand on the bell.
“Five,” she said.
“Correct!” said Coach Harder.
Stephanie winked at Catherine and mouthed “thanks” to her.
“What is the smallest whole number that uses the letter a in its English spelling?” asked Coach Harder.
Joe hit the bell as he yelled out, “One thousand!”
“That’s correct.”
The score was tied again. And on it went throughout the round. We’d get one right, then Joe would answer one. Sometimes Joe was on top; sometimes we were in the lead. As the time ticked down for the round, no one could manage to pull ahead. The round ended with both teams still tied.
“Since both teams are still tied, we’ll now go into sudden death. The first team to get a correct answer will be our winner,” said Coach Harder. “Are both teams ready?”
We nodded nervously. Joe didn’t look as confident as he had when the lightning round started. Coach Harder pulled the next question card, looked at both teams, and then read, “What is the tenth number in the sequence of Fibonacci numbers?”
Catherine’s hand hit the bell in a flash. At the sound of the bell, Joe looked up in shock.
“The answer is fifty-five,” Catherine said confidently.
“That is correct, and the Math Kids are the winners!” Coach Harder said with a smile.
We swarmed around Catherine, hugging her and patting her on the back.
“How did you know that?” I asked in amazement.
“Maybe you haven’t noticed, but fifty-five is my favorite number,” Catherine said.
“It’s our favorite number now, too,” I said.
CHAPTER 16
We were still celebrating on Monday morning as the four Math Kids walked to school. Justin high-fived me about every twenty steps. My hand was starting to get sore by the time we reached the playground, but I didn’t care. There wasn’t anything that was going to ruin my good mood today.
How wrong that thought turned out to be. In the next twenty minutes, my day went from fantastic to absolutely horrible.
It started when we ran into the bullies waiting in the hallway outside our classroom.
“Just because you nerds won some stupid contest doesn’t mean we won’t be waiting for you after school today,” Robbie said threateningly.
“Yeah,” chimed in Sniffy.
“We’re not afraid of you,” said Catherine, although I wanted to tell her to speak for herself, because I was certainly afraid of them.
“We’ll see about that after school,” said Robbie.
“Yeah,” said Sniffy.
We ducked into the classroom before things could get worse, but it wasn’t any better in there, either. Mrs. McDonald was standing in front of Mrs. Gouche’s desk, her finger bobbing up and down as she yelled at our teacher.
“Someone is responsible for tricking my daughter, and a lot of other kids, too, I might add, into showing up for a singing competition that never happened!” yelled Mrs. McDonald.
“But I don’t know anything about it,” said Mrs. Gouche. “I saw the flyers, but I don’t know who put them up.”
“Well, somebody knows, and I’m going to find out who!” Mrs. McDonald shouted as she stormed out of the room.
I had almost forgotten about our trick to get Susie off our team. Mrs. McDonald was like a dog with a bone. She wasn’t going to just let this thing go. So now I had that and the bullies to worry about. Could anything else go wrong today?
Unfortunately, the answer was yes. Our math team was called to the principal’s office after lunch. Mrs. Arnold didn’t waste any time in getting to the reason for us being there.
“Joe Christian believes that your team cheated in the math competition this weekend,” she said.
“What?” I asked in amazement.
“He thinks you knew the questions ahead of time,” she answered. “He and his parents will arrive soon to discuss this, but I wanted to give you a chance to explain before they get here.”
“There’s nothing to explain,” Justin said. Catherine, Stephanie, and I nodded strongly in agreement.
“Okay, please wait by the secretary’s desk until I call for you,” she said sternly.
“This day went from great to lousy pretty quick, didn’t it?” I said when we plopped down onto the chairs.
“Does anyone know what she’s talking about?” asked Stephanie. “We won fair and square.”
We didn’t have long to wait. Joe Christian and his parents entered the office about ten minutes later. The secretary showed them into the principal’s office. After a short time, we were asked to join them.
“Joe believes that the math competition might not have been fair,” Mrs. Arnold started.
“What wasn’t fair?�
�� I asked.
“There’s no way you could have come up with that last answer so quickly without knowing what the question was ahead of time,” Joe accused, pointing his finger at Catherine. “How many other questions did you know about?”
Catherine responded by laughing. Everyone looked at her in amazement, including Stephanie, Justin, and me.
“You think I didn’t know that fifty-five is the tenth number in the Fibonacci series?” she asked Joe.
“No, I don’t,” he said stubbornly. “No one knows that off the top of their head.”
“Oh yeah? Well, what if I told you I do know that off the top of my head? Let me tell you a few other things about the number fifty-five,” she said. “Fifty-five is also a triangular number. It’s the sum of the first ten numbers.”
Everyone in the room was now looking at Catherine, but she was just getting started.
“In fact, fifty-five happens to be the largest Fibonacci number that is also a triangular number,” she added. “It’s also the sum of the squares of the first five numbers. You want more? The number fifty-five is also a Kaprekar number. If you multiply it by itself, you get 3025, and thirty plus twenty-five equals fifty-five.”
The room grew silent. Stephanie, Justin, and I tried to suppress the smiles on our faces, but we weren’t very successful. Mrs. Arnold was also smiling a little. She looked over at Joe’s parents.
“It sounds like Catherine knows quite a bit about the number fifty-five,” she said. “Were there any other questions you thought were unfair?”
Joe’s parents looked at each other and then at Joe, who was taking a sudden interest in his shoes.
“No, I think her explanation was good enough for me,” Mr. Christian said. “Good enough for you, son?”
Joe mumbled a few words under his breath.
“I’m sorry, I didn’t hear you, Joe,” Mrs. Arnold said.
“Yeah, that’s good,” Joe said, a little louder this time.
Joe and his parents rose. His parents thanked the principal for her time and left quickly. Mrs. Arnold smiled broadly after they were gone.
“That’s very impressive, Catherine,” Mrs. Arnold said.
“Well, it is my favorite number,” Catherine said with pride.
“I can tell,” the principal responded.
I rose to return to class.
“Not so fast, Jordan,” said Mrs. Arnold, the smile now gone from her face.
“Yes, ma’am?”
“There’s the other matter of a singing contest. You wouldn’t know anything about that, would you?”
The look on my face was all the confession she needed.
“And the rest of you were in on this, too?” Mrs. Arnold asked.
“Catherine didn’t know,” Stephanie said. “It was all our idea. We wanted her to be on our team instead of Susie.”
“Do you know how upset Susie is? Did you even consider that?” asked Mrs. Arnold.
We all looked down, feeling ashamed of how we had fooled Susie. Mrs. Arnold let us suffer for a long moment, then burst into laughter.
“I wish I could have seen the look on her face,” she said.
When we began to laugh, she silenced us with a stern look.
“And I’ll deny ever saying that. Now get back to class!” she said.
On the way back to class, I felt very relieved, but I knew we weren’t out of the woods yet.
“Two problems down, and one to go,” I said. We still had the bullies waiting on us at the end of the day.
“I’m not worried about them,” Catherine said defiantly. I wished I had her confidence.
At the end of the day, we followed Catherine out to the playground. As they had promised, the bullies were there. Robbie, Sniffy, Bill, and Bryce stood in a line blocking our path.
“Well, nerds, what now?” said Robbie.
“You tell us—you’re the ones in our way,” Catherine replied.
“We got detention for that little stunt you pulled with the spitball,” Bill said.
“Yeah,” said Sniffy, adding nothing to the conversation as usual.
“So?” Stephanie challenged.
“So, you’re going to pay for it,” Robbie said.
“We weren’t afraid of kidnappers, so why should we be afraid of you?” Catherine said firmly.
Robbie’s face began to turn red. He clenched his fists at his sides. He moved toward Catherine, but Stephanie, Justin, and I stepped in front of her. Sniffy, Bill, and Bryce were on the move, too, surrounding us on all sides.
“FBI! Everybody freeze!” came a loud voice from the edge of the playground.
Everybody froze in place, but our heads all turned toward the voice. I was shocked to see Special Agent Carlson walking toward us with his FBI credentials held out in front of him.
“You four, over here!” he barked to the Math Kids.
Then he turned to the four bullies. For a long moment, he didn’t say a word—just examined them with a disgusted look on his face.
“What do you have to say for yourself?” he asked Robbie.
Robbie stammered, trying to get out a couple words, but finally said they weren’t doing anything wrong.
“Is that right, Robert?” Agent Carlson asked.
Robbie looked surprised that the FBI agent knew his name. That look of surprise turned to one of shock at the agent’s next question.
“Are you aware of the penalty for pulling a fire alarm, young man?”
Robbie’s face went white. “Please, whatever you do, please don’t tell my dad,” he pleaded.
The agent gave Robbie a long look, but he wasn’t done yet. He turned his attention to Sniffy.
“As for you, Brian, I’m sure you know it is a felony to set off explosive devices inside a school building, don’t you?”
Sniffy didn’t say anything—just stared down at the ground. He gave a big sniff as he tried to hold back tears.
“I’m going to give you a break today, kids. I’m not going to recommend prosecution of these serious offenses—yet.” The agent paused to let the seriousness of his words sink in. “But if I hear one word from my juvenile undercover team that you are threatening anyone at this school, I’ll be happy to bring in my team to make arrests. Is that understood?”
The bullies nodded.
“I said, is that understood?” the agent asked again firmly.
The boys responded in a loud chorus, “Yes sir.”
“Good. Now get out of here!” Agent Carlson boomed. The boys scattered, running from the playground as fast as their legs could carry them.
Special Agent Carlson turned to us with a smile.
“Do you really have a juvenile undercover team?” Justin asked in awe.
“I do now,” he replied.
“We’re going to be part of the FBI?” I asked.
“Not officially, but those bullies don’t need to know that, do they?” he answered. “And, unofficially, I could use your help on an ongoing case I have.”
“Are you serious?” Justin asked.
“Totally serious. One of the clues looks like it involves some math that I think you might be able to help me with,” the agent said. “If you are up to the challenge, that is.”
“I think I can speak for all of us, Agent Carlson,” I said, looking at Justin, Stephanie, and Catherine for acknowledgement. “We’re in!”
“Great. Come down to the FBI office tomorrow after school and I’ll go over the case with you,” he said. “By the way, how did your math contest go?” Special Agent Carlson asked, although I had a strong feeling he already knew the answer.
“We won!” Catherine yelled.
“Good job. Oh, that reminds me, I have something for all of you.”
We followed him to his car, where he pulled a cardboard box from his trunk. He opened the box and pulled out four jerseys. They were red and black, matching our school colors. Each jersey had Math Kids printed across the front and each of our names on the back, curving over the jersey nu
mber.
Catherine’s jersey number was 55, of course.
APPENDIX
FACTORIALS
Factorials are used to find the number of ways we can order elements in a set. For example, how many ways can we arrange the numbers in the set {1, 2, 3}?
1, 2, 3
1, 3, 2
2, 1, 3
2, 3, 1
3, 1, 2
3, 2, 1
For three elements in a set, there are six different ways we can arrange them. We can show this using the equation 3! = 6.
A factorial is written by putting an exclamation point after the number. To calculate a factorial, we just multiply all the numbers from 1 to the number:
3! = 3 × 2 × 1 = 6
4! = 4 × 3 × 2 × 1 = 24
5! = 5 × 4 × 3 × 2 × 1 = 120
Factorials get big really fast. For example, 5! = 120 but 10! = 3,628,800.
There is one tricky thing about factorials. You would probably think that 0! would be equal to 0, but it’s actually equal to 1. One reason for this is that there is only one way to arrange a set with no elements.
So just memorize this one: 0! = 1.
It was this tricky part of factorials that Catherine knew, which allowed Justin to solve The Sixes Problem for the number 0.
(0! + 0! + 0!)! = (1 + 1 + 1)! = 3! = 3 × 2 × 1 = 6
PIGEONHOLE PRINCIPLE
Imagine there is a big storm and 10 pigeons try to get out of the rain. If there are only 9 holes, at least 1 hole will have 2 pigeons in it.
The pigeonhole principle—sometimes called Dirichlet’s Box Principle after the mathematician Peter Gustav Lejeune Dirichlet—seems very simple, but it is really very powerful.
There is another version of the pigeonhole principle that says the biggest number in a group must be at least as big as the average number. Let’s say we have 5 pigeons and only 2 pigeonholes. The average number of pigeons in a hole is 5 divided by 2, or 2½. Since the biggest number of pigeons in a hole must be at least 2½, that means that there must be at least 3 pigeons in 1 hole, since we can’t have half of a pigeon in a hole.
Here’s another example of how we can use this principle: If there are 6,000 American students at a college, at least 120 of them must be from the same state. How do we know? The average number of students from each state is 6,000 ÷ 50 = 120. The pigeonhole principle then tells us there must be at least 120 from the same state since the maximum number of students from one state must be at least as large as the average.