### Brain Teasers

# Cornered

When Nora and Corman stopped by the school to whisk Monette off to lunch, the poor principal was hemmed in by two tall, annoyed math students.

"My answer is right, and I want my three points!" the blond one cried.

"But it's not the answer Mr. Collins left on the key!"

"He's wrong," the other one insisted. "My answer is right!"

Corman leaned over to whisper in Nora's ear, "Does she need our help?"

"Yours, probably. Mr. Collins is at a conference for the weekend, and Monette took his math class. It looks like she handed back an exam."

"I teach math. Can I help?"

All three talked at once; Monette was as frustrated as the two students. While they jabbered, Corman managed to read the problem in question:

"My house is built on a strangely-shaped lot, so my back yard is a right triangle with an area of 840 square feet. Each of the three sides measures an integral number of feet. If I want to stake my dog so that he can reach the entire yard, how long must his rope be?"

Monette finally pulled rank on the adolescents. "It's not really your problem, but I'd be glad of some help. Mr. Collins marked their papers wrong; their answers don't match his. The students *sound* right, but math was too long ago for me; I'm not sure. To make matters worse, they don't even agree with each other."

"I see," said Corman, playing for time while he mentally checked the answer on the key in Monette's trembling hand. "That's the answer Mr. Collins left you?"

"Yes," she began.

"The right answer is 8 feet shorter," the blond claimed.

"No, it's a lot longer," insisted the other.

Corman held up a hand for peace. "Actually, you *could* all be correct."

"How can there be ... oh, it's one of those problems with an infinite number of right answers, because we're missing a clue?"

"Well, let me see your work." The students handed him their papers; he followed the algebra, nodding for almost a minute.

"Nice work. You've found the only three right answers."

"How can there be exactly three? Usually, it's one, two, or infinity?"

"Usually, you're right. Welcome to discrete math. When you restrict things to whole numbers sometimes they don't work out so simply."

How much rope did Mr. Collins really need?

"My answer is right, and I want my three points!" the blond one cried.

"But it's not the answer Mr. Collins left on the key!"

"He's wrong," the other one insisted. "My answer is right!"

Corman leaned over to whisper in Nora's ear, "Does she need our help?"

"Yours, probably. Mr. Collins is at a conference for the weekend, and Monette took his math class. It looks like she handed back an exam."

"I teach math. Can I help?"

All three talked at once; Monette was as frustrated as the two students. While they jabbered, Corman managed to read the problem in question:

"My house is built on a strangely-shaped lot, so my back yard is a right triangle with an area of 840 square feet. Each of the three sides measures an integral number of feet. If I want to stake my dog so that he can reach the entire yard, how long must his rope be?"

Monette finally pulled rank on the adolescents. "It's not really your problem, but I'd be glad of some help. Mr. Collins marked their papers wrong; their answers don't match his. The students *sound* right, but math was too long ago for me; I'm not sure. To make matters worse, they don't even agree with each other."

"I see," said Corman, playing for time while he mentally checked the answer on the key in Monette's trembling hand. "That's the answer Mr. Collins left you?"

"Yes," she began.

"The right answer is 8 feet shorter," the blond claimed.

"No, it's a lot longer," insisted the other.

Corman held up a hand for peace. "Actually, you *could* all be correct."

"How can there be ... oh, it's one of those problems with an infinite number of right answers, because we're missing a clue?"

"Well, let me see your work." The students handed him their papers; he followed the algebra, nodding for almost a minute.

"Nice work. You've found the only three right answers."

"How can there be exactly three? Usually, it's one, two, or infinity?"

"Usually, you're right. Welcome to discrete math. When you restrict things to whole numbers sometimes they don't work out so simply."

How much rope did Mr. Collins really need?

### Hint

Mr. Collins's answer key gives the actual dimensions of the yard; he didn't realize there were two other answers from the hints he gave.### Answer

37 feet.There are three triangles that fit the given hints, having sides

40, 42, 58

24, 70, 74

15, 112, 113

From the students' comments, we know that the Collins yard has the middle-sized hypotenuse.

A right triangle inscribes neatly into a semicircle, with the hypotenuse forming a diameter. The shortest rope needed to reach all points is one staked at the midpoint of the hypotenuse, forming a radius of the circle. 74/2 = 37 feet.

The blond student found the first answer above, and would have used a rope of 29 feet. The other student figured it at 113/2, or 56.5 feet.

Historical note:

Lewis Carroll searched for the answer to the basic mathematical problem, sent to him by a reader. He found two right triangles with the same area, but couldn't discover a set of three. Somewhat ironically, Carroll had found 20-21-29 and 12-35-37. Had he doubled their dimensions, he would have had the first two triangles above, and quite likely would have found the third.

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