Beth Malmskog

Math etc.

Month: May, 2012

Your super-hip firework-fuse cake-timing adventure.

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This whole story was just an excuse to post a picture of fireworks, one of my favorite things. Photo by John Turner, via Wikipedia.

Ah, the joy of the hyphen.  I just learned a lot about compound words and I stand by the hyphenation of this title.  Because this story is about an adventure, which is of a timing nature, specifically the timing of baking a cake, and the timing is accomplished by fuses, specifically those for fireworks, and the fuses are hip, in a way that I would describe as super.  Here’s the story:  You–I ask you to imagine yourself in this adventure–have decided to learn the amazingly hip art of making artisan fuses.  You spin them on some kind of excellent gravity powered spindle, out of the fiber of rooftop-grown cotton, with many essential oils and secret compounds incorporated.  You are supposed to mix all the junk up together, then spin it, and the amount of stuff you have will determine how long it takes the fuse to burn.  So you can measure out the material to make a fuse that will burn for exactly an hour, or exactly two hours, or whatever you want.  However, the fuse that you create will be very non-uniform.  It will burn at a rate that is not perfectly correlated with the length of the fuse.  If you cut an hour long fuse in half, one half might take 10 minutes to burn and the other half 50 minutes.  There’s just no way to tell.

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No, not this kind of Cake.

So, you have plenty of fuses laying around your house.  You decide that you’d like to bake a cake, perhaps to celebrate some amazing math friend’s birthday.  The cake needs to bake for exactly 45 minutes.  Oh darn.  You realize that you have no clock or timer of any kind in your house.  You impulsively threw them all away in an effort to free yourself from THE MAN.  This hasn’t been a problem until now.  With a jolt, you realize that fuses could make excellent timers.  After all, they are each designed to burn for a specific amount of time, right?  You look around the house and find that you have plenty of 30-minute fuses and plenty of 1-hour fuses, but no 45-minute fuses.  This does not seem particularly helpful, though, since you need to time out exactly 45 minutes.  However, I claim that this is actually not a problem.  The question is this–how can you use a 30-minute fuse and a 1-hour fuse to time out exactly 45 minutes?  No cutting is involved.

If you know the answer, you still have time to win a t-shirt!  Just send your answer to mathmostly@gmail.com.  This is your last chance!  Eeek!

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Last week’s puzzle also required you to split the difference.  In that case, you needed to use the fact that the two containers were cylinders.  The key point is that if a cylinder is tilted diagonally so that the syrup just touches the brim and the point where the wall meets the floor, it will be half full.  So, use that technique to get exactly half of each of the containers.  That is, get 3 cups of syrup in the 6 cup container and 2 cups of syrup in the 4 cup container.  Then, pour syrup from the 6 cup container into the 4 cup container until it is full.  This will take 2 of the 3 cups, leaving you with exactly one cup in the 6 cup container.  Brilliant.

Congratulations to Emmie Finkel and Justin Goldman who won this week’s t-shirts!

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Maple Syrup Fiasco

Well, I’m a little late with this week’s installment.  This is properly last week’s installment.  But it has been crazy around here!  Seriously!  Again, I have to link to the most appropriate song for the end of every semester as a math professor–The Final Countdown.  Thank you Europe. If you, you in all your big-haired glory, only knew the terrible unrealistic air guitar that has been played to your song by the mathematically stressed, well, I think you would be a little surprised.  And hopefully proud.  The song is most appropriate in combinatorial situations, but I try to fit it in to my life ever’ damn semester.

Ah, Europe. Yes, count me down. Finally.

I will be getting back to blogging about things other than puzzles in a couple of weeks, after Math Mostly runs its last episode this Friday.  Only one more puzzle, and no waiting for an answer!  It is sure to be a very special episode of Math Mostly, filled with tears and maybe a montage of the best moments of the semester.  Plus I will announce the winner of the big semester contest. I think we’ve got it figured out, unless there are some last minute answers from one close contender…  So it’ll be great!  However, I was just trying to share a little news about something mathy other than the show.  My recent paper, with Rachel Pries and Bob Guralnick, was just accepted to the Journal of Algebra!  That’s awesome!  I am so excited.  This is likely to be the last paper directly out of my thesis projects, so I’d better get working on some new math.  Or work harder on what I’ve started anyway.  So watch out, math, I’m coming to get you.

Yes, this story takes place in the spring but who can pass up colored leaves?

 

So now the puzzle!  This week’s puzzle involves my co-host Jed and his maple syrup farming uncle.  It was a bad year for maple syrup, I hear, and the farmers have my sympathy.  Jed visited his uncle a few weeks ago and found him with a dilemma.  The uncle some containers on hand that he usually used to give his maple syrup gifts to the family.  There are two sizes–one that holds exactly 4 cups and one that holds exactly 6 cups.  They are perfect cylinders–circular, right cylinders.  And no, I’m not going to tell you the diameters or heights, because you don’t need that information to solve the problem.  Which is this: Jed’s uncle has calculated that since it was such a bad year, he can only afford to give gifts of 1 cup of syrup to each of his family members.  Fine, but he doesn’t have any one cup containers.  So he wants to use the old ones, but containing only exactly one cup of syrup. Jed shows up for a visit, and his uncle needs to measure out exactly one cup of syrup for his gift.  How can he do it, using only the existing 4 and 6 cup containers and no other measuring devices?

If you know the answer, send it to mathmostly@gmail.com!  And tune in this Friday to hear the answer, as well as find out who won the big prize.  2:30-3 pm on WESU 88.1 FM Middletown, or wesufm.org anywhere with the internet.

Oh yes, and what about last week’s puzzle?  That was the one involving the hats.  On Pi day.  Okay, so the situation is that all of the people must be wearing red hats.  Why?  There must be at least two red hats because everybody has his hand raised.  However, what if there were only two hats?  Say A and B are wearing red hats, and C is wearing a blue hat.  Then, A would look at B and see that B’s hand was raised, and think–B must see a red hat.  However, the only hats that B can see are A’s hat and C’s hat, which is blue. So A would immediately know that he must have a red hat, and would have stood up right away to claim the prize.  B would have been in the same situation, so they would have had to fight for the tart.  Since nobody acted right away, it must be that nobody is in a position to make such an immediate action–there must in fact be 3 red hats.  So the first person who sees that the lack of action is actually a piece of information will stand up and declare her red-hattedness.  Congratulations to Anwar, the winner of the puzzle.  He won an amazing T-shirt–I will definitely post the design for you soon.