Interesting Cube Problem

If the cube has a volume of 64, what is the area of the green parallelogram? (Assume points I and J are midpoints.)

Go ahead, work it out. Then, go here for a more in depth discussion, including a video explanation. Also, see my very simple solution in the comments on that page. (My Precalculus students should especially take note!)

And, welcome, SAT Math Blog, to the internet! Thanks for pointing us to this great problem and creating the nice diagram above.

Soda Mixing Problem (revisited)

I posted a problem back in December that I never got back to answering. Sorry about that. The problem statement was:

Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?

I have had the worked out solution for a while, just haven’t posted it until now. I’m relatively new with \LaTeX, but I’ve typed up the solution here, if you want all the gory details :-).  And yes, Peekay, you got the right answer!

Really Fun Limit Problem

Here’s a great problem that a student brought to me today. For those who’ve been wanting a ‘problem of the month,’ here you go:

The figure shows a fixed circle C_1 with equation \left(x-1\right)^2+y^2=1 and a shrinking circle C_2 with radius r and center the origin (in red). P is the point (0,r), Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. What happens to R as C_2 shrinks, that is, as r\rightarrow 0^{+}

Who am I? (hint)

I posted the following problem back on December 3. I thought I’d post the solution, but then I decided maybe to just give you a hint. I’ve emboldened each true statement. The other statement in each pair is false. I did a lot of trial and error, making lists of numbers and crossing things off, narrowing it down. I didn’t have a great strategy, so see if you can do better. Can you figure out the number now?

There are five true and five false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number.

1a. I have 2 digits
1b. I am even

2a. I contain a “7”
2b. I am prime

3a. I am the product of two consecutive odd integers
3b. I am one more than a perfect square

4a. I am divisible by 11
4b. I am one more than a perfect cube

5a. I am a perfect square
5b. I have 3 digits

Soda Mixing Problem

Here’s a good puzzle for you!

Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?

This problem has been stated in many different ways, with various liquids. I’ve phrased it in my own way. If you search around the internet, you can find the solution. But I want you to think it through. Give an answer and see if you can justify it! (I’ll post my solution later.)

Who Am I?

I’m reposting this great puzzle, found originally at The Math Less Traveled blog. Enjoy!

 

There are five true and five false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number.

1a. I have 2 digits
1b. I am even

2a. I contain a “7”
2b. I am prime

3a. I am the product of two consecutive odd integers
3b. I am one more than a perfect square

4a. I am divisible by 11
4b. I am one more than a perfect cube

5a. I am a perfect square
5b. I have 3 digits

Einstein’s Puzzle (Answer)

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.