by David Cole
Charles can’t sit by Brad ✓
Frances can’t sit by Brad ✓
Debbie and Emily can’t sit together ✓
Amy can’t sit by Charles ✓
We had done it. We had solved the problem. Linda’s dinner party was saved. Stephanie was happily eating a cookie to celebrate our victory. Only Justin was silent, a slight frown on his face.
“What’s wrong, Justin?” I asked.
“Mm-hmm,” he responded.
I ate my own cookie and gave him a minute to finish processing his thoughts. I didn’t even try to guess what he was thinking this time.
“It’s not right,” he said.
“It is right,” Stephanie insisted. “We checked every rule.”
“I know, but it’s not right,” Justin said.
“What’s wrong?” I asked.
“There’s more than one answer to the problem.”
“So?” I asked.
“That’s very un-mathy,” he said.
“Un-mathy?” Stephanie asked. “Is that even a word?”
It probably wasn’t a word. But I understood what Justin was saying. We were used to math having a single answer. 4 plus 4 always equals 8. You would never hear someone saying that 4 plus 4 could also equal 7 or a 143. In fact, that was one of the things I really liked about math. It wasn’t like English where there were lots of words that sounded alike but meant something totally different. That seemed like a real waste (or was that “waist”?) and it just made it harder for me to pass our weekly spelling test.
“I don’t care if it’s a word or not,” Justin snapped. “Math doesn’t work that way.”
That’s when Stephanie stepped in to save the day.
“Actually, it does,” she said.
“What do you mean?” Justin asked.
“What number can you multiply by itself to get 4?”
“That’s easy,” Justin said. “2.”
“Is that it?” Stephanie asked.
Justin thought for a second. “-2 also works,” he said.
Stephanie was right! There were math problems that had more than one answer. I quickly thought of another.
“Yeah, and what about zero times x equals zero?” I asked. “You can put any number in for x and the equation will be true.”
And that’s when he got it. Sometimes there were math problems that had more than one answer.
Justin looked thoughtful for a moment, and then nodded and reached for his glass of milk. He raised it in a toast.
“We did it! To the Math Kids!” he said with a grin.
We clinked our glasses together. We had succeeded in the first challenge of the Math Kids!
Unfortunately, our next challenge didn’t go nearly as well.
Chapter 5
Monday started off on a good note. Over breakfast, my sister told me she owed me one for saving her party. On the way to school I was thinking about all the ways I could cash in on my sister’s debt when I found some real cash, a five-dollar bill blown up against a fence. It looked like it was going to be a lucky day, but I was still shocked to see the 85% written across the top of my spelling test. I was usually happy to get a 65, so an 85 was unheard of for me. I even checked Mrs. Gouche’s math to make sure she had gotten it right. 17 out of 20—yeah, that was 85% alright. I couldn’t wait to show my mom.
To make things even better, Robbie was absent from school. Without him, the class was spared any bullying for the day, since Bill and Bryce usually didn’t start anything without their leader. Apparently Robbie was sick, but I suspected he was faking it. For being the biggest and strongest kid in the class, he sure missed a lot of school due to illness. I’d also decided he was one of the clumsiest kids I’d ever met. Once, he was out for almost a week when he ran his bike off the road and into a gully. He came back to class with his wrist in a sling, but that didn’t stop him from using his one good arm to push the little kids around.
Overall, it was looking like a great Monday.
And that’s when things started to fall apart.
As expected, Stephanie had been put into the yellow math group. That meant the Math Kids had a group to ourselves. Mrs. Gouche was going through some beginning algebra concepts. It was stuff Justin and I already knew, but I think Mrs. Gouche was trying to make sure Stephanie knew it too. Stephanie was getting bored with the lesson and wasn’t making any secret of it. Finally, she came right out and said it.
“Mrs. Gouche, we already know this stuff,” Stephanie began. “Can’t we move on to something else?”
Mrs. Gouche gave Stephanie a look that Justin and I had come to know meant you don’t want to go there, but Stephanie didn’t catch on.
As Mrs. Gouche continued with the lesson, Stephanie interrupted.
“This is baby algebra,” she said. “We were doing this at my school in California last year.”
Mrs. Gouche looked at Stephanie. Justin and I looked at each other. We tried to signal to Stephanie that she should be quiet, but Stephanie had already shown us that she didn’t back down from anything.
I tried to step in to help.
“Actually, this is a good review for me,” I said, hoping to ease the tension.
“See, they already know it too,” Stephanie said. “Can’t you give us something more at our level?”
Justin looked down and shook his head. He knew Stephanie had gone too far, and now we would all pay the price. If there’s one thing we’d learned, it’s that teachers don’t like to be questioned on their teaching methods.
“So, you’re ready for something more difficult?” Mrs. Gouche asked. She asked the question nicely enough, but I could feel something else going on under the surface.
“Yes, ma’am,” Stephanie responded. I could tell she felt pretty proud of herself.
“And you’re sure you can solve more difficult problems?” Mrs. Gouche asked.
Stephanie nodded, but she seemed a little less sure of herself than she had been just a moment before. I think she was a little worried about the determined look on Mrs. Gouche’s face. I know I was.
“Well, let’s see if I can find something at your level,” Mrs. Gouche said.
She walked to the board and drew a picture.
“This is the town of Königsberg,” she explained. “There are four parts of the town, which I labeled A, B, C, and D. There are seven bridges in the town, which I numbered from 1 to 7. Your job is to visit each part of the town, going across every bridge exactly one time.”
“Can we start wherever we want?” Justin asked.
Great question, I thought. Justin always seems to know just the right questions to ask. I usually just jump in and try to solve a puzzle, but Justin likes to do his thinking up front.
“Yes,” Mrs. Gouche answered. “You can start wherever you want and you can go across the bridge from either side. You have to visit all parts of the town and you have to cross every bridge, but once you have crossed a bridge, you can’t go across that same bridge again.”
“Sounds pretty easy,” Stephanie said.
“Think you could have it done by the end of class?” asked Mrs. Gouche.
“Absolutely!” Stephanie said.
“Care to bet a class pizza party on it?” Mrs. Gouche asked with a sly smile.
Before Justin and I could say anything, Stephanie had accepted the challenge. She smiled confidently, but I had a bad feeling about this.
Wait! Do you want to try to solve this puzzle before seeing if the Math Kids can do it?
The old town of Konigsberg has seven bridges. Can you walk through the town, visiting each part of town while crossing each bridge only once?
Good luck
An hour later, we hadn’t made any progress. We came close a couple of times, but the answer remained just out of reach. We tried starting in each se
ction of town, but the result was always the same. Every time, we either missed going over one of the bridges or were forced to recross one.
Mrs. Gouche watched our progress, that sly smile never leaving her face. I was right about that bad feeling.
There was a lot on the line, after all. By now the class had heard that we had bet the class pizza party. If we solved the problem, we would be heroes, with not one, but two pizza parties coming our way. If we failed, though… Well, I didn’t want to think about what would happen if we couldn’t solve it. Even though Robbie wasn’t here, I still didn’t want to see Bill and Bryce waiting for us outside the safety of the school building.
The clock ticked steadily toward three o’clock. As the minute hand approached the top of the hour, we worked faster and faster. We tried each starting point again, going across a different bridge to start. Still nothing. Justin made a chart on the board to make sure we weren’t missing any combinations of starting points and bridges. Still nothing.
Eventually, our time ran out. The bell rang and class was over. The problem lay unsolved on the white board in front of us. We continued to stare at the board as our classmates filed out, many of them throwing wisecracks in our direction as they left.
“Nice job, nerds,” Bill sneered.
“Just wait until Robbie hears about this,” Bryce added, knocking a stack of papers off my desk as he passed.
Finally, it was just the four of us: Stephanie, Justin, Mrs. Gouche, and me.
“Was that problem a little above your level?” Mrs. Gouche asked Stephanie.
“I guess so,” she responded quietly.
“I just can’t believe we couldn’t solve it,” said Justin. “We tried every possible direction from every possible starting point.”
“Are you sure?” Mrs. Gouche asked.
Justin looked carefully at his chart on the board and then nodded. “Yes, I’m sure we tried everything,” he said.
“I meant, are you sure the problem can be solved?” she asked.
The thought had never crossed my mind. A problem with no answer? Had Mrs. Gouche tricked us?
While we thought about her question, Mrs. Gouche packed up her things and got ready to leave.
“Think about this. How many bridges go to area A?” she asked.
We looked. There were three bridges between area A and the other areas.
“What does that mean?” Mrs. Gouche asked.
We looked at her with puzzled looks on our faces.
“Let me ask that in a different way,” she explained. “If you start in area A, where will you end up?”
We looked at the picture carefully. Stephanie was the one who came up with the answer.
“If you start at A, you would have to end up in some other area,” she said. “Cross one bridge to leave, another to come back, and then the third to leave again.”
“Right,” said Mrs. Gauche. “And what happens if you start somewhere else and want to cross every bridge going to area A?”
We looked at the picture again and thought for a moment.
“You would have to end up in area A,” I replied. “One bridge to get there, another to leave, and then the third to get to area A again.”
“And what does that tell you?” Mrs. Gouche asked.
“It means we either need to start or end in area A,” Justin answered.
“Correct,” she nodded. “Now, there are also three bridges going to area C.”
“That means we would either need to start or end in area C,” Justin said, starting to see where Mrs. Gouche was going.
“And what about area D?”
And there it was. Area D also had three bridges, meaning we would have to start or end there too. It was impossible to start and end in three different places. Mrs. Gouche was right, the problem was impossible to solve!
“Please erase the white board when you’re done,” she called back to us as she left the room.
“She knew all along, didn’t she?” Stephanie asked as the sound of Mrs. Gouche’s footsteps faded down the hall.
“Mm-hmm,” Justin said. It was the right answer to Stephanie’s question, but he wasn’t really answering. He was deep in thought once again. I couldn’t even guess what he was thinking about. Over the weekend, we had solved a challenge with more than one answer. Now Mrs. Gouche had given us a problem that didn’t even have an answer.
“We should have thought about it before betting the pizza party on it,” Justin said, not even trying to disguise the irritation in his voice.
“I wouldn’t have accepted the challenge if I didn’t think I could do it,” Stephanie snapped back.
“That’s your problem, isn’t it?”
“What’s my problem?”
“You didn’t think about the team, only about yourself.”
I knew I needed to jump in before the argument got out of control.
“Hey guys, let’s not be too hard on ourselves. That was a tough problem,” I said.
“Not nearly as tough as the one we’ll face tomorrow when Robbie gets back,” Justin said.
Stephanie was deep in thought, nervously playing with the end of her ponytail.
Chapter 6
Justin was right. It did get worse when Robbie came back on Tuesday.
“WHAT DO YOU MEAN THE PIZZA PARTY IS GONE?” he yelled. The other bullies shrank back when they saw how angry Robbie was.
“It’s the new girl’s fault!” said Bill angrily, pointing his finger in Stephanie’s direction.
“That’s it! No more playing mister nice guy,” Robbie growled as he walked toward Stephanie.
Mrs. Gouche entered the room and called the class to order. That quieted things down, but not for long. As Robbie went to the back of the room to hang up his backpack, he “accidentally” knocked all the papers off my desk. As I bent down to pick them up, he whispered harshly into my ear. “You and your nerdy friends have had it!”
“Had what?” I asked innocently, prompting Robbie to plant a huge dusty footprint on my social studies assignment. He might have been doing me a favor. Maybe the dirt would cover up some of the grammatical mistakes and misspelled words on the paper I had written on the Louisiana Purchase.
As Robbie hung up his backpack, he “accidentally” knocked Justin’s off the hook, sending it crashing to the floor with a loud thud. Mrs. Gouche looked up from her desk but didn’t say anything.
Justin’s backpack is always filled to the top. He always has three or four books, but he also carries a wide assortment of other things, depending on what he’s thinking about when he packs it. One day there may be a Frisbee, because he’s trying to figure out what makes them fly so well. On another day, it may be his Rubik’s Cube (he’s working on his own solution but still has to rely on an instruction book to solve it). There could be a ball of string, or a harmonica, or the pair of x-ray glasses he bought at the magic shop so he could see who was knocking on his bedroom door (they didn’t work, of course, but he still put them into his backpack sometimes). Once he even had two bricks, and when I asked him, he couldn’t remember why they were in there. I sort of wished that the backpack had fallen on Robbie’s foot, since Justin probably had something heavy in it.
As Justin, Stephanie, and I sat together in the cafeteria at lunch, Robbie and the other bullies decided to join us. Lunch immediately went from break time to torture time.
“What are you eating, shorty?” Robbie taunted Justin. “Did your mommy pack you something good?”
Justin started to reply, then thought better of it and just continued to eat his sandwich in silence. He was probably hoping that Robbie didn’t see that his mother had cut the crusts off the bread. There would be no end to the teasing if they noticed that.
“Yum, cookies!” Bryce said as he snatched a couple of Oreos from my lunchbox. Like Justin, I decided to just l
et it slide. Sometimes the best way to handle bullies is by not letting them know they’re getting to you.
But sometimes they keep it up anyway.
“What’s the deal with the pizza party?” Robbie asked Stephanie.
“Easy come, easy go,” she said, looking directly into his face. “You wouldn’t have had the pizza party in the first place if it wasn’t for me, so you’re no worse off now, are you?”
Robbie’s face began to turn red, and we braced for his response. He took a couple of deep breaths and rose from the table. When he was standing, he towered over us. He bent over the table and stared intently at Stephanie.
“You better find a way to get that party back,” he threatened. He picked up her milk carton and slowly poured it into her lunchbox, soaking her sandwich, Goldfish crackers, and carrot sticks.
“Or else!” he added as he and the other bullies stomped away.
“Well, that was a pleasant lunch,” Stephanie said as she watched the gang dump their trash and head out to the playground.
“You and I definitely have different definitions of pleasant,” I said.
For the rest of the week and into the next, the tension between Stephanie and Mrs. Gouche continued to build. The teacher barked at Stephanie several times, and Stephanie barked right back. The result was twice as much homework as usual. Normally I liked to do math homework, but Mrs. Gouche was really piling it on. I also noticed that it was a lot more difficult than usual.
Justin had finally had enough. When Mrs. Gouche went over to work with the blue math group, he glared at Stephanie.
“What are you trying to do?” he asked in a harsh whisper.
“Well, she was wrong,” Stephanie replied. “There are at least two better ways to solve that problem.”
“And you’re always right, aren’t you?” Justin said snidely.
Between the bullies, Mrs. Gouche, and Justin and Stephanie arguing, I was glad to hear the three o’clock bell ring. I’d had enough school for one day. Stephanie didn’t walk home with us. I saw her up ahead walking with a couple of girls from her soccer team. Justin and I walked home together, but neither of us had much to say. Our problems at school were beginning to mount, and we didn’t have a good solution to solve any of them.