### Brain Teasers

# No-Fat Frankfurters, A Last Hope

Patty O'Furniture, Kurt, & Rod were walking downtown when they spotted a forlorn friend staring at a small object in her hand. She was propped against a wall, one foot off the ground.

"I know her," Patty said. "She looks like she could use some help. I think she broke a heel off her shoe."

Patty was right; the small object was a 3-inch heel.

"This is my sole sister," the girl groaned, "with the obvious name."

Kurt looked blank. After a few seconds, Rod's face lit up.

"Pleased to meet you, Eileen," he smirked.

"I wish it could be under better circumstances. You wouldn't have any superglue on you? Maybe an extra pair of size-5 sneakers?"

They searched their pockets -- no spare shoes, no modern adhesives. Kurt fell back on his usual solution for all depressing problems: a Rnkfurer.

"Is that a spare heel?" Eileen asked.

"No, it's the world's best snack. Non-fat, low-calorie, high-fiber, made of recycled materials, fortified with five vitamins, and contains 100% of your recommended daily allowance of Public Broadcasting."

"Right," she grumbled, "and it tastes like recycled history essays?"

The three grinned broadly at her. "Try one," Kurt said, proffering one of the shrink-wrapped torpedoes.

Eileen shifted some of her weight back to the toe of the damaged right shoe and tentatively accepted the Rnkfurer. She unwrapped it, took one bite, and saw her ecstatic grin reflected on the faces of the other three.

"Where can I get more of these?"

"We were about to go order more tomorrow. But there's a catch: we always order so one of us gets the quantity that the other has for a percentage."

Eileen got it right away. "So we each get 25 of them, and have 25% of the total."

Rod grinned. "That's cheating. We use different numbers."

"Oh, all right." She thought for a few seconds. "So you can do it a few ways for two people, round off for three, but we couldn't do it for four."

"Really?" Rod wasn't as quick with the enlarged problem. "I'm not surprised, but that means we'll just have to buy in pairs."

"Or change the rules a little," Eileen added.

"How so?" Kurt was ready to do almost anything that would get a Rnkfurer refill.

"Well, how about if Kurt buys twice as many as Rod's percentage, Patty gets twice as many as Kurt's percentage ..."

Kurt looked hopeful, Patty was expectant, and Rod's analytical gears were grinding.

"It's a big order, but we could do it," he finally declared.

The grocer now orders Rnkfurers only in cases 100, but that didn't disturb anything. How many did each student get this time?

"I know her," Patty said. "She looks like she could use some help. I think she broke a heel off her shoe."

Patty was right; the small object was a 3-inch heel.

"This is my sole sister," the girl groaned, "with the obvious name."

Kurt looked blank. After a few seconds, Rod's face lit up.

"Pleased to meet you, Eileen," he smirked.

"I wish it could be under better circumstances. You wouldn't have any superglue on you? Maybe an extra pair of size-5 sneakers?"

They searched their pockets -- no spare shoes, no modern adhesives. Kurt fell back on his usual solution for all depressing problems: a Rnkfurer.

"Is that a spare heel?" Eileen asked.

"No, it's the world's best snack. Non-fat, low-calorie, high-fiber, made of recycled materials, fortified with five vitamins, and contains 100% of your recommended daily allowance of Public Broadcasting."

"Right," she grumbled, "and it tastes like recycled history essays?"

The three grinned broadly at her. "Try one," Kurt said, proffering one of the shrink-wrapped torpedoes.

Eileen shifted some of her weight back to the toe of the damaged right shoe and tentatively accepted the Rnkfurer. She unwrapped it, took one bite, and saw her ecstatic grin reflected on the faces of the other three.

"Where can I get more of these?"

"We were about to go order more tomorrow. But there's a catch: we always order so one of us gets the quantity that the other has for a percentage."

Eileen got it right away. "So we each get 25 of them, and have 25% of the total."

Rod grinned. "That's cheating. We use different numbers."

"Oh, all right." She thought for a few seconds. "So you can do it a few ways for two people, round off for three, but we couldn't do it for four."

"Really?" Rod wasn't as quick with the enlarged problem. "I'm not surprised, but that means we'll just have to buy in pairs."

"Or change the rules a little," Eileen added.

"How so?" Kurt was ready to do almost anything that would get a Rnkfurer refill.

"Well, how about if Kurt buys twice as many as Rod's percentage, Patty gets twice as many as Kurt's percentage ..."

Kurt looked hopeful, Patty was expectant, and Rod's analytical gears were grinding.

"It's a big order, but we could do it," he finally declared.

The grocer now orders Rnkfurers only in cases 100, but that didn't disturb anything. How many did each student get this time?

### Hint

They got a total of 600 Rnkfurers.### Answer

Rod 405, 67.5% ... 67.5 * 2 = 135Kurt 135, 22.5% ... 22.5 * 2 = 45

Patti 45, 7.5% ... 7.5 * 2 = 15

Eileen 15

TOTAL 600

Method:

Let E, P, K, and R stand for the quantity each of them had. Let T be the total, R+K+P+E. The math works much like last time, but with four-hundredths instead of percentages.

Since four times Rod's percentage equals Kurt's quantity, and so on for the other three students, we get

2 * 100 * R/T = K. Rearranging ...

R = K * T/200. Now, for the other students ...

K = P * T/200

P = E * T/200

Now, look at the common factor, T/200. Let's call this F, the factor relating one quantity to the next.

P = E * F

K = P * F = E * F^2

R = K* F = E * F^3

and T = E*(1+F+F^2+F^3)

-------------

Since E, P, and K are each four times a percentage, and those percentages are less than the whole order, E+P+K<200. All the order amounts are integers; an exhaustive search shows that there are no solutions for F<1. Since F=1 is "cheating", F>1, which means K>P>E, and E<200/3.

E+P+K=E*(1+F+F^2)<200, so

F cannot be more than 13;

K is a multiple of a perfect square (numerator of F^2);

E is a multiple of a perfect cube (denominator of F^3); the cube cannot be more than 4 (4*4*4).

Now, a brute-force search of the finite possibilities for F (numerator 1-13, denominator 1-4) and possible multiples that leave E+P+K<200, we end with only one option where the total is a multiple of 100: F=3 and E=15.

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