Six Math Class Activities

Inconsistent Highway Signs

You are driving along a highway and see this sign:
  Nearville 150 miles
  Farville 160 miles

Then, surprisingly, an hour later you see this sign on the same highway:
  Nearville 100 miles
  Farville 109 miles

While appearing inconsistent, the signs are both correct! How can this be possible?

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Farville is 9.6 miles beyond Nearville. The first sign location is 150.0 miles before Nearville and thus 150.0+9.6=159.6 miles before Farville. The second sign is 99.6 miles before Nearville and thus 99.6+9.6=109.2 miles before Farville. So, the signs show the correct rounded distances.

Secret Salaries

A group of 7 employees are out to lunch, sitting around a table, and the topic of salary comes up. They all want to know the average salary of the group, without anyone revealing their own salary. Can you think of a way to do that? [Hint: This can be accomplished without anything being written down.]

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One person adds her salary to some random (and secret) large number. Let's say her salary is $500 per week. She adds, say, $8000 to it and turns to the person next to her and whispers "$8500." This goes around the table with each person adding their salary to the passed number, and then whispering the new sum along. Finally, the original person receives the total. She subtracts the $8000 and divides by the number of people at the table and tells everyone the average.

Eggsactly!

A basket (like, say, an Easter basket) has exactly six eggs in it. Six different people each take one egg. These are normal people and normal chicken eggs. After all 6 eggs have been taken, there is still one egg left in the basket. How could this be? (No one put their egg back in the basket.)

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Our answer: The last person took the basket.
Our favorite teacher-suggestion: There was also a chicken in the basket that laid an egg.

Adapted from a Car-Talk Puzzler

Three pieces of paper are face down, each with a different number written on them (say from minus infinity to plus infinity). Your job is to find the highest number by following these rules:

  • You can pick up a slip at random and either choose that number, or discard the number and pick up a second slip.
  • You can then choose or discard the second number.
  • If you discard the second number, then you automatically choose the last number.

So the question is, can you think of a way to improve on a 1/3 chance of picking the highest number? [Hint: there is a way!]

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Pick up the first slip and discard it. Then pick up the second slip. If the second number is higher than the first number, choose it. If not, choose the third number. In this way you have removed one slip that can't be the largest, so you have improved your chances to 50%. Your students can validate this strategy by running several trials and recording their successes and failures in choosing the largest number.

Missing Dollar Conundrum

Three travelers stop for the night in a hotel. They are charged $30 at the front desk and each contributes $10. Later, a bellhop comes to the room and says there was a mistake; the room price is only $25, so he refunds $5. The travelers each take $1 and give $2 to the bellhop. So, each traveler spent $9 ($27 total), and together they tipped $2, for a total of $29 paid. What happened to the other dollar?

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The question is an obfuscation, implying that a dollar is missing when it is not. Of the original $30, $25 went to the hotel, $3 went back to the travelers, and $2 went to the bellhop.

Making Sense of the Census

A census-taker knocks on a door, and asks the woman inside how many children she has and how old they are. "I have three daughters. Their ages are whole numbers, and the product of the ages is 36," says the mother.

  • "That's not enough information," responds the census-taker.
  • "I'd tell you the sum of their ages, but you'd still be stumped," says the mother.
  • "I wish you would tell me something more."
  • "Okay, my oldest daughter Annie likes dogs."

What are the ages of the three daughters?

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Since the sum cannot help, there must be at least two sets of numbers with the same sum. There is one pair of possible answers that have the same sum: {1, 6, 6} and {2, 2, 9}. Since there is an OLDEST daughter, the answer must be 2, 2, 9.

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