15 degree triangle

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Check out this fun problem at the math challenges blog. I picked it in honor of what we’re doing in Precalculus right now, since it has a trigonometric flavor. Don’t look at the answer until you’ve tried it! In fact, after looking at the answer, I found I had done it a different way. So let your creativity run wild. I’d love to see your proof, if you figure it out. And if you want a hint, come see me or email me.

Should Calculus be on Top?

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Paul Lockhart’s A Mathematician’s Lament.  My response to his essay is posted here, if you haven’t checked it out. Here are some more thoughts on Math Education that pick up where we left off.

I think we agree that we need a structured curriculum that gets students from point A to point B. Of course, our curriculum needs to allow for exploration, discovery, and fun, as Paul Lockhart says. But still, in order to prepare people for the real world, we need to take them from point A to point B. But if we’re honest, only 2% of students (I’m being generous) will be in fields where anything beyond Algebra is required of them. And usually, by point B we mean Calculus or something like that (in RM’s case…it may mean Multivariable Calculus/Diff.Eq. or HL Math). We just accept that point B is always Calculus. But most students will never use Calculus directly. This is a tough thing to come to grips with: It’s absolutely true that 98%++ of students will never apply directly the math we teach. What do you think of that?

Here’s a 3 minute talk by Arthur Benjamin, who suggests we change “point B” to be Statistics, not Calculus, as a response to this very dilemma.

Arthur Benjamin is a professor at Harvey Mudd College and an all-around cool guy. You might enjoy his other, more light-hearted TED talk (performance, really) in which he does “Mathemagic,” found here.

Einstein’s Puzzle (Answer)

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Okay, don’t read any further unless you’ve already tried the puzzle. It’s a classic logic puzzle and can be solved by the standard grid-technique, like commentors suggested. I did the same thing, and I got this answer:

The German owns the fish.

Did you get that too? Here’s the thing: It’s technically not correct, according to a few sources I found. Some people say the correct answer to this problem is “there’s not enough information; the fish isn’t even mentioned in the listed facts.” I’m not sure what I think, but it gives some food for thought. Consider the following new problem and you’ll see why:
Who is the American?
(Fact 1) Winston and Paul are of two different nationalities
(Fact 2) Paul is Canadian.
What do you think? Would you say “Winston” or would you say “Winston could be anything (except Canadian) given the facts”? If you say “Winston” then you’re actually assuming a third fact: One of them is an American.
In the case of Einstein’s Puzzle, we technically need a 16th fact: One of them owns a fish.
So what do you think? Those of us who solved the puzzle in the classic logic-problem grid-solution way simply assumed that fact and got on with our lives. But what do you think? Do you think we can assume that someone owns the fish even though it’s technically not a “fact?” It’s an interesting issue–perhaps just a linguistic one. Let me know what you think.

Einstein’s Puzzle

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Thanks to Drew for pointing me to this fun logic puzzle.

Supposedly Einstein proposed this puzzle and it’s often referred to as “Einstein’s Challenge.” The question is, “Who Own’s the Fish?” And here’s the information. Go ahead and try it. And by presenting this puzzle, I am NOT advocating smoking or drinking, I merely wanted to quote the puzzle as it’s classically written.

Here’s what we know:

  1. There are five houses in five different colors.
  2. In each house lives a person of a different nationality.
  3. These five owners drink a certain beverage, smoke a certain brand of cigarette and keep a certain pet.
  4. No owners have the same pet, smoke the same brand of cigarette or drink the same drink.

Here are the clues:

  1. The Brit lives in the red house.
  2. The Swede keeps dogs as pets.
  3. The Dane drinks tea.
  4. The green house is on the immediate left of the white house.
  5. The green house owner drinks coffee.
  6. The person who smokes Pall Mall rears birds.
  7. The owner of the yellow house smokes Dunhills.
  8. The man living in the house right in the center drinks milk.
  9. The Norwegian lives in the first house.
  10. The man who smokes Blends lives next to the one who keeps cats.
  11. The man who keeps horses lives next to the man who smokes Dunhills.
  12. The owner who smokes Bluemasters drinks beer.
  13. The German smokes Princes.
  14. The Norwegian lives next to the blue house.
  15. The man who smokes Blends has a neighbour who drinks water.

Again, the question is: WHO OWNS THE FISH?

The fact that Einstein proposed the problem is urban legend. I’m not sure why it’s named for him. Also, most sources say something like “2% of people who try this problem get it correct,” which is a completely made up statistic, as far as I know.

 I didn’t get it right, if that’s a comfort to you :-). So see if you can do better than me!