Leila Z and the Terrible Triplets

By Greg Crowther

Illustrated by Steve N. Hewitt

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Text copyright 2013 by Greg Crowther

Illustrations copyright 2013 by Steve N. Hewitt

ISBN: 978-1-304-34499-1

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for Leila

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Leila Z* was a tall girl with greenish blue eyes and

brownish blonde hair that she sometimes wore in braids.

She lived in Oklahoma, near Tulsa, with her parents and

her three brothers.

*Pronounced “LEE-luh ZEE”

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One summer Leila started a cake-baking business so that

she would have more money for buying Legos. Her

customers loved her cakes, but she had little time for

baking once school resumed in the fall. She decided to

change her business into a cake consulting service so that

she could help others with their cakes.

Before long, Leila became known as an excellent

consultant.

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Every day or two,

someone would come to

her house with a

question about cake --

maybe something like,

“My recipe says to

heat the oven to 180

degrees Celsius.

What is that in

degrees Fahrenheit?”

“Just a minute,” Leila

would say while

whipping out her

notebook. “180 times 9 .

. . divided by 5 . . . plus

32 . . . is 356 degrees

Fahrenheit. That will be five cents, please!”

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Sometimes the questions were harder. One warm

afternoon in October, a man appeared at her door, all

sweaty and out of breath.

“Help!” he cried. “I have a kitchen emergency!” He pulled

out a notecard with a recipe on it. “This recipe is for a

cake 10 inches in diameter and three inches high,” he

said. “But I need to make it 15 inches in diameter and

four inches high. I have no idea how much of the

ingredients I will need!”

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“Don’t panic,” said Leila soothingly. “This will just take a

minute. Did you say the recipe is for a 10 by 3, and you

need to make it 15 by 4?”

“Yes, that’s right,” said the man, impressed by her

attentiveness.

Out came Leila’s notebook.

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“Well,” she said, “your cake is in the shape of a cylinder.

The volume of a cylinder equals the surface area times

the height, and the surface area is about 3.14 times the

square of the radius. So, using the numbers you gave

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me....” She spent a minute writing out the calculations.

“...Your cake has three times the volume of the cake in the

recipe, so you’ll need to triple the amounts listed in the

recipe.”

“OK, great!” shouted the man. Then he suddenly looked

worried again. “Wait -- that means I have to multiply

fractions!” he wailed. “What’s three quarters of a cup

times 3???”

Leila turned her notebook toward the man, and she

talked as she wrote. “The number 3 can be written as 3

over 1,” she said. “If you multiply the numerators and

multiply the denominators, you get 9 over 4, which is the

same as 2 and one quarter. Does that make sense?”

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“I think so . . . ” said the man cautiously.

“Good,” said Leila. “Let’s try one more for practice. What

is 1 and one quarter cups times 3?”

“Um, let’s see,” the man stammered as Leila handed him

her notebook. He started to write messily. “5 over 4 . . . 15

over 4 . . . Uh, 3 and three quarters?” he offered.

“Very good!” said Leila.

“All right, I think I understand now. Thank you, Leila!”

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Leila looked at her watch. “That will be 20 cents, please.”

The man threw a one-dollar bill down on the table. “You

can keep the change,” he said. “I’ve got to get back to my

kitchen!”

* * * * * * * * *

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Later that month, Leila faced her hardest problem yet.

When she answered the knock on her door, she saw a

woman standing next to three boys about Leila’s age.

“Hello!” said the woman. “My name is Mrs. Tucker. I’m

hoping that you can give me some advice about my boys,

Teddy, Timmy, and Tommy. Tommy, stop pushing

Timmy!”

“It’s nice to meet you, Mrs. Tucker,” said Leila, “but I

don’t know that much about boys. I mostly help people

with cake.”

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“Actually, this IS about cake,” said Mrs. Tucker. “My sons

are triplets. They were all born on the same day, so we

have a big birthday party for them each year. The

problem is that they always argue over who should get

which piece of cake. Can you help me figure out a way to

keep them all happy this year?”

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Leila thought that this still sounded more like a question

about boys than a question about cake. Nevertheless, she

wanted to help.

“I will try to find a solution for you, Mrs. Tucker,” she

answered. “I’ll need some time to work on this, though.

When is the boys’ birthday?”

“Their party is in one week, on the 3rd,” said Mrs. Tucker.

She handed Leila a card with her phone number on it.

“Please call me if you think of anything! OK, boys, let’s go.

Timmy, stop pinching Tommy!”

* * * * * * * * *

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Leila started thinking about Mrs. Tucker’s problem right

away.

“It might help to have a separate cake just for the

triplets,” she thought. “Each of them could have a nice big

piece. But how could we be sure that each boy would get

his favorite piece? Especially since there always are

different decorations on different parts of the cake...”

The next day Leila talked to her math teacher, Mrs.

Hatchpag. Whenever she got stuck on a problem in Mrs.

Hatchpag’s class, Mrs. Hatchpag always seemed to know

what to do.

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“That’s a tough one!” said Mrs. Hatchpag after Leila had

told her about the Tucker triplets. “Maybe you should

start with a simpler problem. What if there were only two

brothers instead of three?”

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The next day, Leila told Mrs. Hatchpag that she had a

solution to the two-brother version of the problem. “Have

one brother cut the cake into two pieces that he thinks are

equally good, and let the other brother choose his piece

first,” she proposed. “Then both brothers should be

happy.”

“That sounds like a good solution,” Mrs. Hatchpag agreed.

“Can something like that be done with three brothers?”

Leila said that she would find out.

* * * * * * * * *

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Another day went by before Leila had an idea for the

three-brother version of the problem. Here’s what she

thought. A long knife could be moved slowly across the

cake until one of the brothers said that they wanted the

piece that would be cut by the knife in its current

position. That brother would get what he considered to be

a fair third of the cake. The rest of the cake could then be

divided with the two-brother method that Leila had

already discussed with Mrs. Hatchpag.

Just to be safe, Leila decided to test her idea with her own

brothers. She baked a cake, gathered her brothers around

it, explained the rules, and got out a long knife.

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When the knife got about a third of the way across the

cake, Murray claimed the first piece for himself. Then

David took the knife and made the next cut. Because

David was the youngest, and not feeling very hungry, he

made one of the remaining pieces much smaller than the

other. Paul grabbed the really large piece, and David was

left with the really small piece. Both Paul and David were

happy with this outcome. Murray, however, was not.

“Paul’s piece is way better than mine!” he whined. “I want

a do-over!” And Leila realized that her idea wasn’t quite

as good as she had thought.

* * * * * * * * *

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For the next two days, Leila tried to think of a better

solution -- a solution in which all of the pieces would be

decided at the same time, so that there would not be any

surprises.

At last she emerged from her room with a final answer.

She called Mrs. Tucker and told her to bake the triplets a

cake in the shape of a triangle. “I’ll show you how to cut

the cake at the party,” she promised.

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On the day of the party, Leila went to the Tuckers’ house

early and inspected the triangular cake. Then she made

three copies of a picture of the cake as seen from above.

She gave the pictures to Teddy, Timmy, and Tommy, and

told each of them to go to a separate room and make a

preference diagram like this:

The boys were uncooperative at first.

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“Why do we have to do this?” Teddy inquired. “Is this part

of a secret plan to stop us from fighting?” Tommy asked.

But with Leila’s encouragement, they eventually made

their diagrams. Leila collected them and combined them

so that they looked like this:

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Leila took the combined diagrams to Mrs. Tucker.

“This diagram shows the preferences of the three boys,”

she said.

Then Leila added an “X” to the center of the diagram.

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“If you cut from point X to the three corners of the cake,”

Leila said, “Timmy will want the left piece, and Tommy

will want the right piece, and Teddy will want the bottom

piece. All three of them will be happy!”

“As a side note,” she added quietly, “it can be

demonstrated that such a point X exists for the

preferences of any three people.”

Mrs. Tucker was simply delighted. “Wow, Leila -- you’ve

done it!” she exclaimed. “You’ve proved that Teddy,

Timmy, and Tommy can all be happy at the same time!

That’s amazing!”

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After the other party guests arrived, Mrs. Tucker cut the

triplets’ cake from point X, just as Leila had

recommended. Then she brought out a second cake for

the rest of the family and guests. She cut that one in a

more normal fashion. As she finished cutting, she smiled

broadly and announced, “I think Leila should get to

choose the first piece.”

THE END

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Reference:

Edward B. Burger and Michael Starbird (2013). "Cutting Cake for Greedy

People: Deciding How to Slice Up Scarce Resources." Section 10.5 of The Heart of

Mathematics: An invitation to effective thinking (4th Edition). John Wiley &

Sons.

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