Beth Malmskog

Math etc.

Hello Again from the Middletown Inn

I’m out in Connecticut working on some math with the wonderful Chris Rasmussen.  We started this great project when I was working out here and we keep making progress, but somehow the stunning conclusion just keeps getting further away.  It’s like watching Mad Men.  I mean, it seems to me that Don Draper has just got to fully collapse and/or be redeemed one of these days and the story will run out.  But no!  There he is again, drinking in the morning and eyeing another dame.  I keep thinking that these binary forms/s-unit equations/curves of genus 3 are just going to give it up and fall down dead under the weight of their rock and roll lifestyle.  However, they survive and somehow seem to thrive in their folly, suavely dodging all attempts to pin them fully down, somehow gaining my sympathy and drawing me back to Middletown again and again.  I guess this is somewhat less glamourous than a date with Don Draper, though come to think of it, maybe the better bargain.  

This is my way of easing back into this blog after nearly a year off (I am finding it hilarious/pathetic that my last blog entry was entitled “Reintroduction”).  It has of course been a big year and it would be too much to try to say much about it, except I can’t get my head around where all the time went.  I’m deep into my second year at Colorado College (which I love!) and now on my second house in Colorado Springs (I’m really good at moving these days).  Just before this trip I was in Fort Collins visiting Rachel Pries, along with Colin Weir.  Suzuki curve fever.  Also an excellent Halloween.  

Hmm, I was going to post some pictures but my poor computer has too much going on to open iphoto… I’m pushing this six year old computer to its edge and am taking my digital life in my hands every time I use it.  Yes, of course I backed everything up (four years ago).  So pictures will have to wait for next time while my computer cools down a bit.  But now that I’ve taken the edge off I hope to have another post soon.

 

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Reintroduction

This is me, reintroduced to writing and guarding my winter meal with bared fangs.

This is not only a wolf, reintroduced to Yellowstone: this is me, reintroduced to writing and guarding my winter meal with bared fangs. Clearly that elk shouldn’t have made such a ridiculous claim about the cable bill.  Photo from PBS.org.

Wow, it’s time to reintroduce myself to writing.  By which I mean writing about whatever I want to write about.  The last 3 months have been incredibly full of math, speaking, writing on the board, writing lesson plans, emails, website updates for my classes, research and teaching statements, applications–all the standard stuff.  But I haven’t posted for quite a while.  I haven’t sent my energies this way.  However, something truly inspiring happened at my parents’ kitchen table in Laramie over Thanksgiving break, something that revitalized my blogging desire.

I saw a terrible cable commercial.

It claimed you could reduce your cable bill by 150% with their bundle.

I’m guessing that nobody wants to hear the bitter rant that emerged from my until then mild-mannered, turkey-stupored person.  It involved swear words and a lot of contempt.  Come on, you can’t reduce a cable bill by 150% unless the cable company is going to pay you to have cable!! 100% of your bill is your whole freaking bill.  150% of your bill is your whole bill and then half again!  Is this company going to send you a freaking check every month? Seriously!  Does no person in the entire advertising department understand percentages?  Does no person in the entire company understand percentages?  If they do, are they crazy, or do they think that there are actually 0 Americans who both understand percentages and haven’t learned to TIVO everything they could possibly want to watch on television?    Am I all alone in the whole world?!?!

But on the upside:

1) That terrible commercial got me writing about it, which is already fun.

2) In my agitated state I forgot every detail of what they were advertising, so when wanted to find it, I called my mother in Laramie to ask if she remembered.  My 11-year old niece Gabby answered the phone and we talked for a while about her hockey practice, and tennis practice, and when I was coming home for Christmas, and what I should be considering as a Christmas present for her and her brother Nick.  (She said “A science set.”  My heart leapt with joy.  Though she’s the kind of sweet kid that would say that just because she knew it would make my heart leap with joy.) I described the commercial to her and asked her if she’d seen it, and if so could she remember what company it advertised.  She hadn’t seen it but she did ask me all kinds of relevant questions, impressing me with her engagement and problem solving skills.  Then she handed the phone off to my mom, who remembered the rant but not the company.  But while we were chatting I got to hear Gabby in the background, explaining to Nick: “You can’t reduce somebody’s debt by 150% unless you’re going to pay them.  100% is the whole bill.  Nobody’s going to pay you to watch TV.”

3) I searched many variations on “save 150%” to see if I could find the commercial online. I didn’t, and I was reassured that this claim doesn’t seem to appear all over the internet for all kinds of products, as I had feared that it might.  I was getting paranoid about America’s math skills and advertising ethics.  This sold-out item is the only product I found which featured the exact “save 150%” claim.  So, at least on this count, the whole world has not fallen into darkness.

This is the kind of graph you'd better be able to show me if you want to make a claim like that.  What's that you say, cable company?  Nothing?  That's what I thought.  Image credit: Santhanam, et al. ©2012 American Physical Society

This is the kind of graph you’d better be able to show me if you want to make a claim like that. What’s that you say, cable company? Nothing? That’s what I thought. Image credit: Santhanam, et al. ©2012 American Physical Society

4) As usual, I found a lot of things I wasn’t seeking.  My searches brought me to something about “230% efficient light production,” which turned out to be an article about a (bad pun that you will get in a second) “extremely cool” recent feat of physics/engineering: an LED that emits more light energy than it takes in as electrical energy.  It apparently gets the rest of the energy by absorbing heat from its surroundings.  I love this in so many ways!  Claims with percentages that are justified? Yes!  Counterintuitive greater than 100% efficiency that actually makes sense?  Yes!  Energy efficient light production? Yes! It’s December in Colorado and the temperature has been over 60 degrees every day for two weeks.  That’s the forecast weather for the Joint Mathematics Meetings in SAN DIEGO!!  Colorado’s ski industry desperately needs these LEDs.  Though it looks like they currently can only produce very small quantities of light, and that it needs to be 135 degrees celsius for this efficiency to happen.  But still.

5) The cooling lights brought to mind laser cooling, a (note to self: don’t use the cool pun again here!!), a process which won the Nobel prize in Physics and has applications to creating Bose-Einstein condensates (macroscopic quantum phenomena).  I heard about laser cooling when I wrote about Dr. Jacob Roberts work for the school paper during graduate school.  Which leads (because scientists have used lasers to cool/slow particles down enough to trap them for use in quantum computers) to quantum computing.  Something I am especially stoked about because, if a quantum computer is ever built, the kind of cryptosystems that I studied at Microsoft Research will be really useful. In any case, this made me look up the really recent Nobel prize in Physics that was given Serge Haroche and David Wineland for their related contributions to quantum computing.  Which led me to a New York Times Opinion piece about quantum computing, and how great is it that the NYT has opinion pieces about quantum computing?  My search-walk then brought me to a fancy new press release from MIT about very recent advances in the mathematics of quantum computing.  Thanks, Peter Shor and company.  They have proven that the entanglement necessary for quantum computing on a practical scale can be produced in a much simpler situation that researchers feared.  This makes a practical quantum computer much more imaginable.  What if all online transactions were suddenly insecure?  This is what all my back to the land / money in a coffee can / stockpiling guns on their compound friends are worried about (these are three different sets friends, by the way).  EXPLOSIONS!  CRAZY STUFF!

7) Another recent advance which did not win a Nobel prize in Physics that but has applications in my life (this January in San Diego): how to build really tall sandcastles.

This is 10 feet high.  Could they do better after reading this new study?  Public domain photo from Guy King, via Wikipedia.

This is 10 feet high. Could they do better after reading this new study? Public domain photo from Guy King, via Wikipedia.

 

8) In the course of our discussions about this awful commercial, by non-math-loving but very sensible mom pointed out that the ridiculous claim of saving 150% on your cable bill could even make sense if they are considering your cable bundled with your internet.  Idea being that if you look at your current cable and internet bill, you could by bundling with their company get a combined cable and internet bill that is less than your current combined bill by 150% times your current cable bill.  This seems like a pretty weak premise for the claim, it does give the company some room to argue that they are not baldly lying.  So I can to get off my soap box now.  And grade some calculus exams.

Latin square love affair

This is what you get when you google “sexy quilt.” Can you believe it?? My faith in the essential wholesomeness of the internet (which should never have existed in the first place) has not been restored, but I do love surprises.

I am writing today about a problem that I described in my last post about how a group of 5 people could pass 5 quilts around so that each person works on every quilt and no person ever passes to the same other person twice.  And of course, I used Latin squares to model the problem, and I am now a little bit in love with Latin squares.  So the exciting thing for today is my new mathematical crush on Latin squares and a solution to the problem at the end of this post.

But first, let me share a few updates about my life.

School is just about to start here at Colorado College. CC has a very unusual course structure.  The academic year is organized into eight three-and-a-half week blocks.  During each block, students take a single course and professors teach a single course.  Classes are intense but the structure gives the opportunity for lots of creativity in teaching and amazing field trips.  I am really looking forward to teaching on the block schedule.  My first class is good old Calculus I, starting Monday and running through September 24.  So goodbye to all my decadent ways.  Goodbye, sleeping late.  Goodbye, reading novels all afternoon.  Sigh. But really tt’s about time I got back to work.  I’m pretty sure I have the best job around so I’m not going to whine (any more) about not being able to sit home and read The Hunger Games all day.

And… forget work, there is another extracurricular activity to take up my time.  I finally visited KRCC 91.5 FM here in Colorado Springs and had a great time talking with Mike Procell and Vicky Gregor.  Did I mention how much I really love that station?  It was one of the (many) things that made me decide to take the job here at CC.  I will go in tomorrow to sit in with Vicky and start training! The prospect of getting on-air at KRCC is totally thrilling.  So wish me luck.  And, I also get to learn about my health insurance tomorrow.  Did I mention how much I love being out of grad school and having health insurance?

I couldn’t resist sticking in a gratuitous picture of this incredibly beautiful place. I work here!

Here is one of the sweet quilts that Judy and her group worked on, though without the passing sequence they dreamed of.

Back to the problem. In the interest of laziness, I will link directly to a pdf of my solution and response to Judy: quiltproblem.

My favorite part about this problem is that it is a real-life problem that is modeled with very useless-seeming mathematics.  I love abstraction and mathematics for the sake of fun and beauty, and I love solving problems, but its not every day that these two loves come together for me.  And this problem promises to be a good source of united fun for a while.  So, I have determined that the quilts can be passed as desired if there are an even number of people.  It is not possible if there are 3 or 5 people.  What about other odd numbers?  That’s my next question: for what sizes of groups is it possible to find a good way to pass the quilts?

I’ll post whatever I figure out.  But probably not until block break.

Not dead. And a quilt problem.

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Here I am, flying a kite at Province Lands National Seashore, during the 3 months I spent not updating my website.

Well, I have to say that it’s pretty hard to start writing when it’s been three months since the last post.  Eek!  I can’t believe it has been so long.  And did I really not post the solution to the last puzzle?!? I am so ashamed.  But that means there’s only one thing to do–post the damn answer.  Okay, so if you recall, we were trying to cleverly measure out 45 minutes using fuses that burn for exactly one hour and exactly half an hour.  The fuses are non-uniform, and you can’t predict how long one will burn by its length.  If you cut a one-hour fuse in half, for example, and light the pieces, one could burn for 5 minutes and the other for 55.  So what can you do?

Of course the thing to notice here is that a fuse is designed to burn for a set amount of time when you light it at one end.  If you light the fuse at two ends, the fuse will be consumed twice as fast, and therefore in half the originally planned time.  So–light the one-hour fuse at both ends and it will burn for half an hour.  There are now several ways to time out 45 minutes using your one-hour and half-hour fuses.  One thing to do is light one of each type of fuse at one end, and let them burn until the half-hour fuse is consumed.  At this point, light the other end of the one-hour fuse.  It had half an hour left, and now it is burning at two ends, so it will take 15 minutes to burn.  Half an hour plus 15 minutes is 45 minutes.

Alright.  That was the last math mostly puzzle.  I am already missing WESU and the Somewhat Science crew, as well as my excellent co-host Jed Rendleman.  Now that I am all moved in to my new house in Colorado Springs, though, I am making plans to check in with KRCC, the excellent station at Colorado College.  Hopefully they will be interested in letting me do some math on the radio.  I will report back with any news.  In the mean time, I want to share another puzzle of sorts that came to me from a friend.  It involved trading quilt squares in a group of 5 people.  This is a little different from the other puzzles in that it comes from an actual problem that my friend’s mother was having.  It wasn’t designed to have a cute little solution or anything like that.  I did solve it, and it was fun, but it wasn’t entirely simple and cute.  Anyway, I’ll share it and let you think it through.  Here’s the email I got from Judy:

Hi Beth,

My son, Martin gave me your email address. Hopefully he already told you I’m struggling with organizing the group project for some quilters, and can’t seem to get it to work out “exactly” the way I want it to. Here’s the scoop if you feel like a puzzle. I’d certainly appreciate the help and insight.

I am organizing a round robin quilt exchange. That means a group of quilters each make a small quilt block and trade them with each other over a period of time until everyone in the group has had a chance to add a border to everyone else’s quilt. 

On the first trade day, each person passes their quilt block to another person in the group and receives a block from someone else in that group, so they can add a border to that block. After a month or so, the blocks are passed to another person in that group, so they can add a border to the block and so on until everyone in the group has had a chance to sew a border on every block. At that time the block is returned to the person that made the center quilt block.

I’m having trouble setting up the passing order. The last time we did this activity we passed our project to the same person each time until we got our own quilt back. If I passed my quilt to “Martha” then every time we traded, I gave my quilt to “Martha,” which means she always had to follow my work in this game. It is kind of limiting, since part of the fun of doing these trades, is getting to meet the other people in the group. 

I’d like to arrange the trades so no one ever gets their quilt from the same person, but I can’t quite figure it out. I tried it with 5  people in a group and, while everyone gets the quilt to sew on, the quilts are passed twice to the same person. I’m hoping there is an equation to figure this out, instead of my trial by error method.

I think we will pretty much always have five people in a group, but it is possible that we would have groups with 4 or 6 people. Will a formula work for that, too?

Thanks, 

Judy Gilmore

Judy and I both thought to organize our work on the problem in a table, which made the problem into sort of a Latin square with conditions.  A Latin square is essentially a Sudoku puzzle.  The idea was to make a 5 by 5 grid.  Each row in the grid would correspond to a quilt square.  The people would be numbered 1 through 5.  A row would describe the path of the quilt square through the 5 people in the group.  If we could successfully fill in the grid so that each row and column contains the numbers 1 through 5 this would give a way to pass the quilts so that each person worked on each quilt and everybody always had a quilt to work on.  There are lots of ways to do this–for example

12345

23451

34512

45123

51234

The problem is that this grid describes the situation of Judy’s first quilt passing experiment–where each person just passes to the person next to them until the quilts have seen every person in the group.  But that’s not what we want.  How do we translate the condition that each person should pass to a different person each time into a condition on the Latin square?  Well, one person passing to another corresponds to a sequence of two numbers in a row (like if 23 appears in a row that means person 2 passed a quilt to person 3).  So if we never want to repeat passes, we have to create a Latin square in which no sequence of two numbers is repeated in different rows.

Okay, so I will tell you that I either found such a Latin square or proved it was impossible.  And I addressed the problems of 4 and 6 people.  But I’m out of time for now and don’t want to spoil the fun in case someone wants to try it out for themselves.  For now I’ll leave you hanging.

This seems like a reasonable place to put in a plug for Martin Gilmore, my friend and Judy’s son, who is a really fabulous singer/songwriter/guitar player based in Denver.  Besides his solo work, he plays with the bluegrass band Long Road Home and the rock band Bimarinal.  And besides the fun I had with this puzzle, Martin has brought me hours of fun as a guest on Live@Lunch (back in my KRFC days).

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!

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.

 

Pi Day Hat Game Three Way Stand Off!

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Pi pie.

This week’s puzzle is about a Pi party gone horribly wrong.  Okay, not really horribly, but pies were ruined, and the remaining desert became the prize in a hair-raising battle of logical thought.  It all starts with a Pi Day (3/14, that is) dinner party at my friend Hannah’s house.  She is a compulsive knitter, by the way, and is constantly knitting stocking caps.  She made a bunch of amazing pi-themed dishes and invited me and two other friends (call them Alice and Charlie) over for dinner.  We were all feeling happy and math-tastic after dinner when Hannah went into the kitchen to get desert.  Hannah had made the most beautiful little rhubarb-pi tarts for desert.  Having had Hannah’s baking before, Alice, Charlie and I were really excited about these tarts.  So we were dismayed when we heard a clattering sound and a splat from the kitchen, followed by Hannah’s dismayed cry of “Oh ****! I dropped the tarts!”

Three of the tarts were destroyed!  On the kitchen floor, face down, currently being gobbled by Hannah’s dog.  The final tart was shaken but unharmed.  Hannah came in from the kitchen, bearing said tart, and said, “This one tart will never divide four ways.  That’s like one bite each.  No way.  We’ll have to play a game to decide who gets it.”

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It was just like this.

Hannah decided that just the three guests would compete–she would make herself some tarts another time.  So Alice, Charlie and I looked at each other with growing coldness.  Our eyes shifted back and forth, just like we were the Good, the Bad, and the Ugly.  This was serious.  Hannah disappeared into her knitting room and came back with a big box of hats.  Strangely, they were all red or blue, and there were a ton of each color.  She told us all to hold still, and then came up behind us in turn and put a hat on each of our heads.  We couldn’t see our own hats, but we could each see the other people’s hats.  Hannah explained the rules.

“Each of you is wearing either a blue hat or a red hat.  When I say go, raise your hand if you see a red hat.  When you figure out the color of your own hat, stand up.  The first person to figure out the color of her hat wins.  Okay, now GO!”

We all raised our hands.  Then nothing happened for a while.  We stared at each other, and stared some more, hands in the air.  Things were getting tense when suddenly, I realized that I did know my hat color!    I hadn’t known at first but all that waiting had told me something.  I jumped into the air, correctly announced the color, and snatched my delicious prize.

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I found this on the internet and I will be visiting geek knit very soon to order one. Don't tell Hannah.

The puzzle this week is–what color was my hat, and how did I know?

Send your answer to mathmostly@gmail.com!  Best answer wins a math mostly/somewhat science T-shirt.  The shirts are on order and are coming really soon!  See what I’ve been doing when I should be proving theorems.  Also, we’ve got another contest going, where the awesome mystery prize goes to whoever sends in the most correct puzzle answers over the whole semester.  Only about 4 weeks to go, so keep them coming.

 

 

 

Ah yes, and now for last week’s puzzle.  I love the tournament puzzle.  The question–how many games need to be played in a single-elimination, March madness style tournament with 10,041 teams?  By single elimination I mean that a team is eliminated after one loss, and one winner is wanted at the end of the tournament.  The answer is–10,040 games.  The way to see this really quickly is to notice that each game has one loser.  We want one winner, so the other 10,040 people have to lose.  Thus we need 10,040 losers, so 10,040 games.  Congratulations to last week’s winner, Hanako!

What is the most important single-elimination tournament of March? The Rubik’s Cube-Off, of course.

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Surely you’ve heard of the Rubik’s Cube-Off, the annual speed solving cube competition?  You know, the one that takes place on Pi Day, i.e. March 14, and that you qualify for by reciting as many digits of pi as you can? No?  Well, that’s because my frequent math mostly co-host, Jed Rendleman, hasn’t reported on it yet.  Or it could be because it doesn’t exist.  Jed is one of my many excellent students this semester at Wesleyan.  He’s been working with me on the radio versions of the last few puzzles and is really great at asking questions and being funny and all the other stuff that makes for a wildly successful math radio show.  Anyway, this weeks puzzle story is about Jed’s spring break and how I helped him cheat at the cube-off.  Which of course, just to be clear in case Jed is applying for grad school (or someone is background checking me!) didn’t actually happen.  No wire was worn into a no-communication-allowed tournament.  No windowless van was employed.  No digits of pi were recited.  But it does make for some kind of puzzle. 

So, the story is that Jed, being a journalist, wanted to go behind the scenes to report on the Cube-Off for the Wesleyan Argus.  He decided to enter as a contestant to get a real taste of the action at this, the bloodiest and most contentious Rubik’s Cube competition in the world.  The qualifying round, which consisted of reciting at least 120 digits of pi in under 1 minute, was not problem for Jed.  Being a bit of a savant, he had no problem memorizing the digits in a very short time, and handily qualified for the tournament.  He took his place among 10,040 other pi stars hoping to be the one winner of the Cube-Off.  So how does this competition work, you ask?  Well, first of all, beyond the qualifying round, it has nothing to do with pi.  Except they serve pie at the winner’s banquet, in celebration of the hallowed date.  The tournament itself is just your basic single elimination tournament, in which contestants go head-to-head in many rounds of cube-solving.  In each match, two contestants are given scrambled cubes. They compete to solve the cube as quickly as possible.   The one who solves it faster stays in the running, while the slower solver is out of the tournament.  

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Early idea for choosing U.S. President--head-to-head Rubik's Cube competition.

Because of the nature of the qualifying round, every year a different number of people qualify to compete in the tournament.  That means the organizers can’t plan out the structure in advance.  They don’t even know how many matches to schedule.  So they have to figure it all out very quickly after the qualifying round, so that the Cube-Off can begin.  Alright, so the first question is, how many matches are there in the tournament?  The tournament judges decide to offer a cash prize to the first contestant who can tell them.  

Here’s where the ethics violation comes in.  You see, Jed was afraid that he would blow his cover at the Cube-Off because it turns out that he is not really nerdy enough to be at one of these things.  He doesn’t know any Star Trek jokes, has no idea how many sides the D&D dice have, can’t extensively discuss the merits of the old TI-85 versus the new TI-82, let alone take a side in the TI versus HP debate.  That last is all about graphing calculators, by the way.  Hopefully you, like Jed, have no idea what I’m talking about.  But the Cubers do!  Jed was afraid that the other contestants would find him out as a fake nerd and he would be a tournament outcast before he had a chance to gather enough material for his story.  So he’d asked me to be a sort of nerd coach, on call in case he needed some explanation or tips in passing for the kind of person who would be into all that stuff.  We decided that he should wear a wire and that I should be on call outside, in a van, ready to help in emergencies.  This was totally illegal, because outside help, like all calculating devices, are banned from the tournament.  They don’t want anybody programming some kind of cube solver into their TI-85s.  

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Aw, just like my first graphing calculator! So cute!

Okay, so this is getting a little out of hand long, so I will get to the point.  Jed used his wire to ask me how many matches would need to be played in this years tournament of 10,041 people (including Jed) to get a Cube champion.  As a math puzzler, I immediately knew the answer.  I did one incredibly easy mathematical operation in my head and told him how many matches it would take.  He answered, and won the cash prize.  And immediately forgot all about the story he was writing.

So this week’s puzzle is:  How many matches are there in a single elimination tournament of 10,041 people, and what is the insight that let me figure it out incredibly quickly with a single mathematical operation?  If you know the answer, write me an email at mathmostly@gmail.com.  The best submission I get by Wednesday, April 4 will earn its writer an excellent t-shirt!

Solution to the planes puzzle:  Ah!  So, this puzzle got a few answers despite the spring break lull.  Here’s one solution.  You can do it with 3 planes.  Call them A, B, and C.  Airplane A will be the one to fly all the way around the world.  Let’s say it takes 8 hours to fly around the world, just to keep it straight.  A, B, and C take off to fly “clockwise” around the world.  1/8th of the way (or after 1 hour), plane C gives planes A and B each enough fuel to go an additional 1/8th of the way.  Then C turns around and goes back to base.  1 hour later, 1/4 of the way around the world, plane B gives plane A enough gas to go an additional 1/8th of the way around the world, then turns around to go back to base. Plane A is now 1/4 of the way around the world, has a full tank of gas, and 2 hours have passed.  Now, after 2 hours later, when B arrives back at base, B and C fuel up and take off to go around the world in the opposite direction.  They go 1/8th of the way around the world that way, taking one hour, then C gives enough gas to go an additional 1/8th of the way around.  C turns around and goes back to base.  5 hours total have passed.  After another hour passes and B has gone another 1/8th of the way around the world, A meets B.  B gives A enough gas to go 1/8th of the way around (I should have named a unit for this!!), then turns around and proceeds with A back towards base.  6 hours have passed.  At this moment, C leaves base with a full tank of fuel.  After an hour, when 7 hours have passed (and A has made it 7/8ths of the way around the world), C meets A and B and gives them each enough fuel to get back to base.  Everybody gets home safe, at the same time!  

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Hopefully this doesn't make you more confused...

How many hipster airplanes does it take to fly around the world?

Yeah, you probably don’t know, because, um, you’ve probably never heard of these airplanes anyway, because besides being really obscure I also made them up.

It doesn't take this many planes, I'll tell you that.

This week’s puzzle is about airplanes.  The story starts with a team of scientists discovering an amazing new fuel source.  This fuel produces no greenhouse gasses or pollutants of any type.  It is renewable.  One catch is that the only known source of this fuel is a tiny, very obscure island in the middle of the Pacific Ocean.  So the scientists have made their base on this island.  They are getting ready to announce their findings to the world.  Hooray! All of the world’s energy problems are solved.

However, these scientists are so out of touch with reality that they think they actually need a publicity stunt to go along with this announcement.  They decide that they will fly an airplane around the world using this amazing fuel.   One airplane must make a complete, uninterrupted circuit around the earth.  Yes, the long way around.  The pilot must fly a great circle, like the equator (though it could be a different great circle, for example including both the north and south poles).  Anyway, everybody agrees that this is a great idea (heh), but the problem is that when the scientists retrofit the airplanes they’ve got to run on this fuel, each plane can only hold enough fuel to take it half way around the world.

So it seems that the scientists are out of luck.  However, one scientist notices that there may be some hope if they use another plane as a helper.  Perhaps one plane could transfer fuel to another in flight.  They try this out and find that it works!  In fact, the pilots, planes, and fuel are all so amazing that fuel can be transferred from one plane to another in midflight, instantaneously and without any complications.  The two planes don’t even have to be going the same direction.  And another thing about these planes is that they change direction instantaneously, too, so you don’t even need to worry about that.

So everything seems to be back on.  However, the scientists soon realize that one helper plane won’t quite do the job.  To see why, lets consider what happens in the most basic scenario.  We’ll call the plane that is going to hopefully make the whole trip around the world plane A.  The helper plane will be plane B.  Let’s call the units of fuel worlds (Ws).  So we say that each plane can hold at most ½ W of fuel at a time. Considering the situation for a second, you see that our only possible hope is if both planes take off at the same time going the same direction with ½ W each of fuel.  The best that we can do here (without plane B running out of fuel on the way back and crashing into the ocean, which we don’t want) is to have B go 1/6 of the way around the world with A, then transfer 1/6 W of fuel to A, then go back to base with its remaining 1/6 W of fuel.  This will get plane A 4/6 of the way around the world, but if B refuels and takes off to meet A on the other side, B won’t be able to carry enough fuel to get them both home safely.  Ugh.

So the scientists clearly need to get some more planes involved.  The question is, how many planes to they need and how can they do it?

 

If you know the answer, send me an email at mathmostly@gmail.com!  The best answers I receive by Wednesday, March 28, will win amazing math mostly/somewhat science t-shirts for their authors.

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Congratulations to last week’s winners, Brian and Eli!  I got some great answers this week so it was a hard choice.  If you didn’t win last week, but you had the right answer, I’ll remember you and like you more next time.  So keep trying.  Thank you thank you to everyone who wrote in.

The solution to last week’s puzzle is simply this—the two kids go first, taking 2 minutes.  The fastest kid goes back, taking 1 minute.  Grandma and Mom cross, taking 10 minutes.  Pretty fast kid goes back, taking 2 minutes.  Both kids cross over, taking 2 more minutes.  Grand total: 2+1+10+2+2=17 minutes!  Awesome!

Reading the solution over, this may not seem challenging at all.  However, most people only come up with this solution after struggling for a long time trying out scenarios in which the fast kid always runs the flashlight back across the bridge.  This seems like the fastest way, right?  However, we know that this is doomed to failure because it means that each other person must cross with the fast kid, for a total of 17 minutes crossing time just going one way! That leaves the kid no time to ever cross back with the flashlight.

The old perilous-bridge-at-night-with-dying-flashlight puzzle

 

 

The bridge is like this, only, um, scarier. The internet didn't have any pictures as scary as the bridge in the story. And it is dark out.

A new puzzle!  A family of four is on one side of a rickety bridge across a deep and otherwise uncrossable chasm.  They desperately need to get to the other side.  Perhaps they are being pursued by monsters.  I don’t know.  But I do know that it’s night time–totally dark, no moon.  The bridge is a weak thing, which can only support the weight of two people at any given time.  And it doesn’t have a hand rail, or anything like that, so you could walk right off the edge if you can’t see where you’re going.  So anyone who is on the bridge had better have a flashlight with them.  Our family of four has a flashlight.  Just one, though, and the batteries are dying.  In fact, there are only 17 minutes of battery power left.  The family knows that it will take one of them (call her Grandma) 10 minutes to cross the bridge.  Another of them (Mom) will need 5 minutes, another (fast kid) will take 2 minutes, and another (really fast kid) will take 1 minute to cross.  If two people are on the bridge together, they will have to travel at the rate of the slower person, because they have to share the flashlight.  There is no funny stuff, like throwing the flashlight (or throwing Grandma) across.  The question, obviously, is how can the whole family get across in 17 minutes or less?  Can you save this family?

If this family lived in Connecticut, they would still be traumatized by the Halloween storm and would each have been carrying a hand-crank flashlight/radio with them at all times. Along with sleeping bags and some MREs. They could listen to WESU as they crossed the bridge, leisurely, one at a time if they felt like it. Luckily for us, they are poorly prepared.

Figure it out!  Send solutions or questions to mathmostly@gmail.com by Tuesday March 6.  Best solutions will receive an incredible math mostly/somewhat science T-shirt!  Listen to Somewhat Science from 2:30-3:00 Friday afternoon (Eastern time) on WESU 88.1 FM (wesufm.org) for the solution.  And a lot of other great science related stories from Wesleyan students.  Speaking of Wesleyan students, big thanks to Jed Rendleman, my Math Mostly co-host this week.

Solution to last week’s puzzle:  Congratulations to James Ricci and Katherine Mullins, the two winners of last week’s Math Mostly write-in puzzle explosion! The puzzle was called the Trivium’s Checkerboard, but is commonly (and slightly gruesomely) known as the mutilated chessboard problem.  Their solutions are so good that I’ll just quote ’em.

James:  “Milo noticed that the checkerboard alternates between black and white squares. That is, adjacent to any black square there are only white squares, and vice versa.  Therefore each individual domino must cover exactly one white and one black square at once. However, there are exactly 32 black squares and 30 white squares! So no matter how he places the dominos down he will only ever be able to cover at most a set of 30 pairs of black and white squares leaving at least 2 black squares uncovered.”

Katherine: “White squares are only adjacent to black squares on Trivium’s board, and vice versa.  Thus, any domino that covers two adjacent squares must cover one white square and one black square.  Two white squares were removed from the checkerboard, so Trivium’s board has 2 more black squares than it has white squares.  If the only dominos available necessarily cover one white square and one black square, then at least 2 black squares remain uncovered at all times.”