Language Students Problem

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This is a problem from the 2001 American Invitational Mathematics Exam (AIME).

“Each of the 2000 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let m be the smallest number of students who could study both languages, and let M be the largest number of students who could study both languages. Find M – m.”

Answer.

See Language Students Problem for a solution.

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Smart Money

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This is a simple puzzle from Futility Closet.

“Mr. Smith goes to Atlantic City to gamble for a weekend. To guard against bad luck, he sets a policy at the start: In every game he plays, he’ll bet exactly half the money he has at the time, and he’ll make all his bets at even odds, so he’ll have an equal chance of winning and of losing this amount. In the end he wins the same number of games that he loses. Does he break even?”

Answer.

See Smart Money for solutions.

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Maximizing Love

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This is a Valentine’s Day puzzle from BL’s (Barry Leung) Math Games.

“Happy Valentine’s Day everyone, I hope you are having a euphoric moment, but if not, you can try this algebra puzzle about maximizing the expression LUV + LU + UV + LV given L + U + V = 12, where L, U, V are non-negative integers.”

Answer.

See Maximizing Love for a solution.

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Three Buckets Question

From Presh Talwalkar here is a variation of the three jugs problem.

“You have buckets that hold 3 L, 7 L, and 20 L of water. How can you measure the following amounts?

  1. a) 10 L
  2. b) 4 L
  3. c) 5 L

For most of mathematical history the above information would be sufficient information to state the problem. But in today’s society, there is a demand to state all assumptions as if that will make the problem better. So the test explained there are certain actions you can take.

You can fill any bucket completely with water. You can pour all the water from a bucket into a larger bucket. You can pour water from a bucket to fill a smaller bucket. You can empty the water completely from any bucket.”

See the Three Buckets Question for solutions.

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Interest Calculations

A recent video by Angela Collier about compound interest reminded me of an effort I did years ago to derive the formulas for interest calculations, especially for installment payments on a loan like a mortgage.  In the process I showed where Euler’s constant e can show up.  I thought I would resurrect the material, even though I imagine modern texts now provide the information (I couldn’t easily locate it back then).

See Interest Calculations.

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ZigZag in Rectangle

This is another problem from BL’s Math Games.

“What fraction of the rectangle is colored?  Assume that M and N are midpoints of the sides of the rectangle.”

That they are midpoints was not stated explicitly in the problem as given in front of the subscription wall, but from the comments it became evident this was the case.

Initially I actually assumed the line was positioned arbitrarily.  What would be the solution in that case?

Answer to BL problem.

See ZigZag in Rectangle for a solution.

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ChatGPT Problem

This problem is from BL’s Math Games.

“What’s the area of the red triangle?”

BL decided to see what kind of solution ChatGPT would come up with.  After several tries and prompts it seemed to oblige.  I don’t know what BL’s prompts were, and in the statement of the problem outside the subscription wall he never explicitly says what the problem is, namely, to find the area of the red triangle.

There also seems to be some ambiguity about the constraints on the problem, that is, how much of the appearance of the diagram should the solver assume?

Answer.

See ChatGPT Problem for a solution.

Post Views: 112
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