15 Jul

I’m a huge sucker for deals.  The other day I wanted ice cream and was going to buy a pint, but then saw that for just $1 more I could get TWO half-gallons (a two for the price of one deal). So I ended up getting those, only to realize when I got home that I’d accidentally bought LOW FAT ice cream (egads!) So I feel like I got suckered into buying a product that I didn’t want. That said, we’re still eating the ice cream; it’s just not as good. That story was just an intro to say that this dish is a really good deal. It’s very, very little work for a lot of return-literal “ooh”s and “aah”s. My friend Chris (who, incidentally, made one of my favorite sites involving Rodents Of Unusual Size) sent this to me via one of those chain-email recipe exchange things that never work. Lucky for me I got this recipe, which I’ve made again and again since 2011. You’ll maybe not want to whisk life and limb for this souffle, but definitely try it to your fullest CAPacity I realize I took the above photo a little bit late- the first thing you want to do is put a stick of butter in your pan of choice (Chris likes deep and narrow; I like whatever I have on hand), and stick it in your oven, which you’ll heat to 425. Not a pun: I think I’ll start calling grilled cheese sandwiches “butter melts”. Sounds so good right now. While the butter is melting, put all your other ingredients in a bowl and whisk, whisk away. I do the eggs first, then other stuff. I hate it when people sugar-coat bad news Just give it to me and maybe I”ll figure out if it’s a mixed blessing or not By the time your oven is preheated, your butter should be melty and bubbly. Pull out the pan, pour your bowl of delicious stuff right on top of the butter, and toss it back in the oven. If someone with ombre blonde-brunette hair hangs from her knees on monkey bars, would you call her upside-Brown? This POURcelain dish is perfect for what’s happening here. Who needs root beer floats when you can have a butter float? After thirty minutes, you’ll have a gorgeous, poofy souffle that took none of the work compared to usual souffle recipes. I’ve had it with cinnamon sugar on top, maple syrup, or my favorite, toss on cut up berries (any kind) and a spoonful of sugar- the heat will slightly cook the berries and sugar. Serve IMMEDIATELY (or the poof will go down) with a side of berries. I love the almost crispy chewy crust in contrast with the creamy center pieces. It’s all delicious. Chris’s French Toast Souffle 1/4 cup (1 stick) butter 3 eggs 1.5 cups milk 6 TB sugar 3/4 cup flour 1/4 teaspoon salt Optional toppings: more sugar, cinnamon, maple syrup, berries… Put the stick of butter in a pan, place in oven, and preheat oven to 425. Whisk eggs in a bowl. Add milk, sugar, flour, and salt and mix until smooth. If the butter isn’t bubbling yet, wait. When it is, pour batter into the pan over butter. Bake for 30 minutes. Serve immediately with desired toppings. ## The fault in “The Fault in Our Stars” (Cantor’s diagonalization argument) 8 Jul I love The Fault in Our Stars- I’ve read the John Green book three times and my husband and I have discussed at length how much we like it. We just watched the movie last weekend and thoroughly enjoyed it (not as much as the book, but it’s a tough act to follow). There’s just one little thing wrong in it-a bit of math! Both my officemate and my fairy blogmother have posted on social media about a flaw in something that Hazel says, so I thought I’d take a post to explain both a) why Hazel is wrong and b) how the argument that she refers to works. For the record, I believe that c) it’s fine for teenagers in books to not fully understand these sorts of mathematical arguments, and it’s unclear whether John Green believes that the infinity between [0,1] is the same “size” as the infinity between [0,2] or not. This is true and is what we will show, but Hazel says (beautifully) in the book: There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities… I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful. Makes me tear up a little every time. And not because the math is wrong, but because the writing is so pretty. But the math is, in fact, wrong. All those infinite sets are the same size. But she is right that some infinities are bigger than others. Speaking of math, I’ll do just a little bit more reminiscing before we dive in. During my sophomore year of college, I took Set Theory with the other two math and philosophy majors (some of my now-best friends) and we saw this argument for the first time. Blew. My. Mind. This is also one of my two cocktail party math tricks (I’ve already written about my other one). I’m not that fun at cocktail parties. So. Let’s talk about sizes, shall we? Let’s first talk about finite counting. How do I know that I have three potatoes? I look at the potatoes and I assign a number to each one. These potatoes are hiding their sadness- not enough eyes compared to their peers. Another way to say this is that I have written a bijection between my set of potatoes and the set {1,2,3}. That is, a function from one set to the other so that each item in set A gets assigned to exactly one item in set B (1), and so that each item in set B has someone assigned to her (2). Property 1 makes a function injective, and property 2 makes a function surjective. If your function is both injective and surjective, we say that it’s bijective. One way to imagine a function is lining up all the items of your sets next to each other, and drawing a back-and-forth arrow between the items of the set. These po’ taters… these smiles are hiding so much pain And we can do this for any finite set. Click photo for credit. I see a bijection to the set {1,…,14}, so there are 14 bunnies. So far so good, this is just normal counting. But what about counting to infinity? Is there more than one kind of infinity? (Spoiler alert: yes). How could we know? To deal with infinity, we extend how we count from finite sets. We say that two sets are the same size if there exists a bijection between them. For instance, the set of bunnies and the set of potatoes are not the same size, because there is no bijection between them. I can see that by trying to make a bijection between A={1,2,3} and B={1,2,3,4,5,6,…14}. No matter how I assign a number to 1, 2, and 3, there are still 11 numbers in set B that aren’t assigned a number in set A. So I can’t have a surjection, therefore I can’t have a bijection, therefore A and B are different sizes, ergo the size of the bunny set is not the same size as the potato set. Technical aside: if there is a bijection between sets A and B, and another one between sets B and C, you can put them together and get a bijection between sets A and C. This jives with our intuition that if A and B are the same size, and B and C are the same size, then so are A and C. “Being the same size” is an example of an equivalence class, a math term that we may use in the future (but not right now, so no formal definition just yet). Here’s a first example of playing with infinity. I claim that the set of natural numbers {1,2,3…} is the same size as the set of even natural numbers {2,4,6…}. Proof by (completely unnecessary) picture Eye just like putting eyes on things This is our first idea of infinity. In fact, we use this infinity so much that we have a word for when a set is the same size as the natural numbers: we say that set is countable. Here’s the next question to ask: are all infinite sets countable? Is there an infinite set which isn’t countable? Or as Hazel puts it, are there some infinities that are bigger than others? Here’s where Georg Cantor’s ingenious argument (from the 1870s) comes in. Let’s try to see if the numbers between 0 and 1 are countable. We’ll do so the same way we tried to see if there were the same number of bunnies and potatoes: start building a bijection, and come up with a contradiction. Fact: every number between 0 and 1 can be written uniquely as an infinite decimal expansion. By infinite, here, we mean countable. Examples: 1/100=0.010000000…. 1/3= 0.33333333….. 1/11=0.09090909090…. So let’s start building our bijection by putting the natural numbers on the left, and infinite decimal expansions on the right. I started with these examples here: Assume I have finished building my bijection and have a list of all the natural numbers on the left, and the numbers between 0 and 1 on the right. We’re going to build a number between 0 and 1 that doesn’t show up on this list. This proves that our assignment function isn’t surjective, and therefore isn’t a bijection. And our argument will hold for any way that you try to build a bijection, and so it’ll show that there are in fact no bijections between the two sets. How do I build the number? Let’s look at the first coordinate in its decimal expansion. I look at the first number in my list, 0.01000000, and look at its first coordinate: it’s a “0″. So we’ll assign any digit besides 0 to the first coordinate in our special number. Let’s say it’s 1. So far, my number is 0.1_________. Now let’s look at the second coordinate. I look at the second number on my list, 0.3333333333, and its second coordinate: it’s a “3″. So let’s make our second coordinate a 4. Now my number is 0.14______. Third: I get 0.141________________ I keep going. To make my ith coordinate, I look at the ith number in my list, and the ith coordinate of that number. Then I pick any digit which is not that one, and assign it to my ith coordinate. Now let’s say we’ve finished building our special number coordinate by coordinate. Is it somewhere in our list? Think about it for a second… …. Answer: no! If our special number were on our list, it’d have to be assigned to some natural number n. But by the very way we built our special number, its nth coordinate is not the same as the nth coordinate of the number assigned to n. So it can’t be on the list. So our special number, which by construction lies between 0 and 1, is not in the image of our function. So our function isn’t a surjection, and hence it isn’t a bijection. This means that we do have more than one infinity: the infinity in [0,1] is not the same size as the infinity in {1,2,3….}. Some infinities are bigger than others. (we finished goal b) But some aren’t. (goal a). In particular, the bijection we did earlier between the sets {1,2,3…} and {2,4,6…} works to show that the infinity between 0 and 1 is the same size as the infinity between 0 and 2. That is, $x \mapsto 2x$ is both injective and surjective. Also, $x\mapsto 1000000x$ is also a bijection, so the infinity in [0,1] is the same size as the infinity in [0,1000000]. Sorry Hazel, you’re wrong in your example (but the idea is correct!). I hope you’re enjoying a little bit of infinity in your day! (Actually, a lot of infinity. Or at least, uncountably much.) ## The best burger ever 2 Jul I went through a big burger phase a few weeks ago (we may have had four burgers in three days), and this recipe really is the best burger ever. It was better than the$14 burger at the fancy butcher shop across the street.  It’s better than any burger I’ve had.

Don’t get me wrong, I love a good barbecue and grill marks and all that (in fact I had my first grilled burger of the summer last night!).  But you just can’t get the same juiciness on a grill as you can in your cast-iron (because that juiciness will just drip down the grill and away from your burger).  Honestly I’m not sure how much the “smashing” step does for the crust of the burger (wouldn’t throwing a patty into a searing hot cast iron sear it just as well?), but I do it anyway because this recipe has worked out so well for me.  It’s also made me realize why people follow recipes closely- because they work, over and over again!

After we got that cast-iron for Christmas, I started cooking a lot more meat.  By now I’ve made these burgers three or four times and they’re delicious every time.  I also got a meat grinder attachment for my KitchenAid (thanks in-laws!), so the first few times I ground my own meat (half chuck, half sirloin).  But we live across the street from an excellent butcher and their fresh-ground meat is just as good.  However, if you don’t have access to fresh-ground meat and just see the stuff in the store, I highly recommend seeking it out.

Yeah, I work out. That’s how I got such nice-looking buns.

I love ketchup, but these burgers are so good that ketchup would just distract you from the flavor-all you need is that melty cheese and sweet grilled onion (pressed right into the patty), and maaaaybe a slice of tomato/lettuce.  Honestly I put on the tomato just to please my husband (because then there’s a bit of health on the plate).

Anyways, let’s start with meat grinding.  If you’re using the one I used, you’ll want to slice your meat into strips (maybe 1-2 inches wide), and then throw it in the freezer while you put together your machine/do something else for awhile.  Don’t forget when cutting chuck to AVOID the white ligament-y parts (they’ll get stuck in the grinder and be a hassle).  Another reason to use sirloin (which is delicious!)

I was really nervous to meat his parents

I thought it’d be a terrible grind

Apparently if you want it to be as finely ground as at the store, you should grind your meat twice.  I didn’t do that and it was still delicious.

Next, start heating up that cast-iron skillet on medium-high or high if you like to live dangerously.  The cast-iron skillet is key.  Incidentally, the original website I got this from uses a big green egg for cooking, and a friend of ours has a crazy amazing website all about the Big Green Egg if you’re into that.  I’m very impressed by it.

While that’s heating up, slice your onion up (not the smart way) laterally so you get some rings, and separate those out.  Take out your cheese of choice.  Defrost your buns.  Slice your tomato.  Then make some meatballs!  Each of mine was 1/4-1/3 lb.

I was afraid my jokes would be too cheesy, I’d turn red as a tomato, I’d make someone cry, or flip out, and/or all of the above. But my in-laws think I’m the greatest thing since sliced bread!

Now put two or three meatballs in your cast iron and let cook for 30 seconds (I used my microwave timer).  Reset your timer for 2 minutes.  Smash down the meatballs (I need a metal spatula!), and press some onion rings into each one.  Then generously toss on some salt and pepper (THIS IS ALL THE SEASONING YOU NEED- use good meat!).

I shouldn’t have worried- neither of us has ever really gone ball-istic in stressful situations.

Overall it was a smashing success!

I just had to remember to be myself, not un-Yen.

Now’s a good time to think about toasting those buns.  I thought the next step would be hard, but it was actually super easy- flip the burgers so that they land on the onions.  You can smash them down a little to hold the onions there.  Let that cook for another two minutes, then put on a slice of cheese (if you want).  If you don’t want the cheese, you should still do the next step: cover with a lid and let cook for one more minute.

Cheese Louise I’m done with the in-law stories, I promise

You can put a lid on your complaining already

All of my worries have melted away

I generally do a double stack of these for a meal, and a single stack for a snack (yes I’ve impulsively stopped at the butcher, bought a half pound of chuck, and made burger snacks for the two of us at 3 p.m.  I also bought two slices of cheese from the butcher).  You can always make more if people want them- it only takes 5.5 minutes from start to finish.

Best burger ever: link here (they also have better pictures than me)

Incidentals: this is my 100th blog post!  Huzzah!

I am currently in Somerville, MA doing this super cool research program.  What this means is that I don’t have access to a lot of my usual baking tools, or central air.  So we’ll see how the posts go for the next several weeks (maybe I’ll be more mathy!).  I am planning on making a rhubarb pie since the postdoc who was living with us for a week made a delicious and beautiful one twice.

## Two things I tell calculus students (one is the squeeze theorem)

22 Jun

I was subbing for a friend in our math tutoring center the other day and ended up chatting with an undergraduate who was retaking calculus.  She asked if I was a grad student in math, and when I affirmed, she said “wow, you must have memorized so many formulas.”  I laughed.  I told her that math is a lot like cooking.  Yes, you do need to memorize a few basics (how to cut an onion, general measurements like tsp to a TB, etc.), but you certainly don’t need to memorize every recipe you’re going to use.  You should definitely read them through and understand the rough idea of what’s going to happen; the more recipes you read, the better you’ll know how to use various ingredients.  And if you just pick up a cookbook and read a random recipe, maybe you’ll branch out to more exotic ingredients and figure out yourself how to incorporate rutabaga into your existing repertoire.

Hilarious photo from coursera (click for link)

To make the analogy very clear: you should read and understand formulas, proofs, etc. very well, but no one expects you to be a walking textbook.  For a single class or a single exam, yes, you should know the info there.  But the idea is that from studying a theorem really hard for a while, you’ll remember the key idea for much longer than a semester.   Logic is hard, proofs are hard, math is hard.  You have to work really hard at the basics before you can make a perfect souffle.

Another way this analogy works: no one learns to cook by memorizing cookbooks.  You learn to cook by getting your hands dirty in the kitchen, trying out random recipes from the internet, and burning a few more complicated things that you weren’t ready for.  If you’ve never chopped vegetables with your dad in the kitchen as a kid, sure, you start at a disadvantage, but that doesn’t mean you can’t pick up a knife and try.  Use youtube videos, ask friends, cook with friends!  Now replace all the times I said “cook” in this paragraph with “math” and pretend that math is a verb.

Check out this cookbook! There are similar math books

Students (like me) often think we won’t cut it in grad school because we don’t have the experiences of others- didn’t do undergraduate research, take graduate courses while in undergraduate, maybe didn’t even major in math.  But just because you didn’t help your parent as a kid doesn’t mean you can’t cook now, and just because you didn’t focus on math before doesn’t mean you can’t do it now.  You learn to do math by doing math.

So that’s the first thing I tell calculus students, or at least this one that I was talking to last week.

Second thing: she asked me to explain the squeeze theorem to her.  Will do!  My explanation of it involves an old family curse.

So when my little brother was born, someone who was mad at my parents cursed our family.  Luckily they weren’t too mad, so it was a pretty benign curse: I would always be shorter than or the same height as my oldest brother, and I would always be taller than or the same height as my baby brother.  (Another way to say this: I’ll never be taller than my big brother, and my little brother will never be taller than me.  This affects our sibling basketball games, but that’s about as bad as the curse gets).

I think this is pretty good considering I googled ‘basketball’ and ‘basketball hoop’

We grow up, and we always grow according to the curse.  One day when we’re grown ups, someone sees my two brothers and realizes that they’re the same height.  Without even seeing me, they can answer: How tall am I?

… (this is you thinking)…

Yup, I’m the same height as those two!  This is the squeeze theorem, because my brothers’ heights has squeezed mine.

Replace our heights with functions: let’s say my brother’s names are Gerard and Hugo, and indicate their heights at time by g(x) and h(x), respectively.  And I’m f(x).  Since Hugo is always taller than or the same height as me, we have an inequality: $latex f(x) \leq h(x)$ for all time x.  Similarly, $latex g(x) \leq f(x)$.  Putting these two together, we have $g(x) \leq f(x) \leq h(x)$.

The squeeze theorem says that if for some where all three functions have a limit, $\displaystyle \lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L$, then we have forced ourselves into $\lim_{x\to a} f(x) = L$, just as Gerard and Hugo’s heights forced mine to be the same as theirs.

Two things I tell calculus students!  I actually tell calculus students a lot of things (like calculus not using family curses), but these are the two things I told a calculus student last week.

## Snickerdoodles, or more morning baking

12 Jun

This KitchenAid stand mixer is changing my life!  Baking during breakfast is awesome!

No normal subgroups here! Sorry, this is a math pun, and only if you’ve taken abstract algebra (no normal subgroups = simple)

I particularly liked my students this semester, so I made cookies for the last day of class.  The first summer I taught (the very first college course I taught!), I made two kinds of cupcakes for my students.  But I didn’t actually like the students that much (the cupcakes were DELICIOUS- margarita flavored (tequila-lime cupcake with salted lime frosting) and Irish car bomb flavored (Guiness cupcake, filled with whiskey-chocolate ganache, topped with Bailey’s frosting)), just baking.  And it was summer, so I had extra time.  Anyways.  These cookies were thrown together while I was making hashbrowns + eggs for

If I put some trinkets in a glass of lemon+sugar+water, it probably won’t be as good a kitsch in ade as this one!

Snickerdoodles are ridiculously straightforward: beat butter, sugar, eggs, add flour and leavening.  Pretty much all snickerdoodle recipes out there tell you to use cream of tartar+baking soda instead of baking powder, but I didn’t have any.  Whatever!  They still ended up delicious!  Form into balls and roll in cinnamon-sugar.

I should try this pick up line on husband while making these: will you be a sin o’ mine, sugar?

We watched Clueless yesterday and I can’t stop thinking about “rollin with the homies”

Bake for not too long while you eat breakfast!

Sometimes he likes to help out in the kitchen, but he never knows when I’ll go BALListic on him (those are too close, damnit!)

These also cool nicely and pack well in tupperware.

Ridiculously easy snickerdoodles (adapted from the internet and cut in half from most recipes, makes about 2 dozen)

1 stick butter (1/2 c butter)

3/4 c sugar

1 egg

1 1/2 c flour

1/2 TB baking powder

1 tsp cinnamon

1 TB sugar

Beat the butter with the sugar until fluffy, then beat in the egg.  Add flour and baking powder and beat.  Form into balls.

Mix the sugar and cinnamon in a bowl.  Roll the snickerdoodle balls in the cinnamon sugar, then place about 1 inch apart on a baking sheet (they do spread, but only a little bit)

Bake at 400 for 10 minutes until just golden brown on the edges (you want crisp edges but soft and cakey middles).  Let cool.

## Cantaloupe-mint sorbet, or we got an ice cream maker!

6 Jun

Speaking of gifts, husband got me some flowers as a welcome-home present from that conference I went to in Ann Arbor (more on that later).

This is the least flour-y photo ever put on a baking post in this blog

I loved the arrangement, and I especially loved that they put fresh mint into it!  Immediately after getting me from the train station we went grocery shopping, and I picked up a cantaloupe and a lemon.  I came home to that new ice cream maker, and knew what had to happen.

Sugar and water might be a great match, but their parents are traditionalists- they can’t elope.

Sometimes it’s hard for me to tell if I’m joking, or if I mint every word in my captions

Sorbet is actually pretty easy: make a sugar syrup (or use honey or agave), and blend it with your fruit+ flavors.  I boiled a cup of water with a cup of sugar, then let it simmer while I cut up a cantaloupe.

Mmm… melons

He’s water. And boy, he’ll (BO-IL)

be excited to see his SWEETheart.

Let’s give them some privacy to mix

I have yet to figure out a reasonable way to cut up a melon.  I just sort of hack it into chunks.  That’s fine for this, because you just throw it in a food processor/blender with the juice of a lemon and some mint leaves until it’s pureed.

Water might start getting jealous- cantaloupe is looking very (c)hunky!

Did ya hear there was a breakout at the Spooner grove? Jemons on the loose! Err… Lemons on the juice!

Then you mix in enough sugar syrup to taste.  I made a cup of simple syrup, but only used about 3/4 of it.  You want to make the mixture a little bit sweeter than you like, since freezing it makes it less sweet (ice crystals… science… stuff…).

I sometimes wish I’d been born two weeks early so I could be a Torus, sign of the bowl (wow that doesn’t make any sense! Worst bowl ever!)

Then throw it in your ice cream maker and let it run for 20 minutes or so!  I’ve never used one before so it was pretty amazing to see it go from complete liquid to this thickened substance stuff.

Everything was mixed up earlier. Glad we got it SOR(be)TED out.

By the time this was done, our steak was resting and salad on the table, so I scooped it into a tupperware and put it in the freezer until we were ready to eat it.  I love the brightness of the mint and lemon popping out of the sweet taste of cantaloupe (which I evidently love from that smoothie post).

Missed the sorbet this time…

Cantaloupe-mint sorbet, adapted to be an easier version of this Better Homes and Gardens recipe:

1 cantaloupe

1 c sugar

1 lemon

5 sprigs of mint

Bring 1 c of water + 1 c sugar to a boil.  Reduce heat to low and let simmer until thickened while you dice the cantaloupe, leaf the mint, and slice the lemon.

Turn off the heat under the simple syrup, and if you have electric, move the pot to let the syrup cool.  Meanwhile, puree the cantaloupe, mint, and lemon in a food processor (you will probably have to do this in batches) until smooth.  Then add cooled sugar syrup to taste (somewhere between 1/2 c and 1 c, depending on your melon).

Refrigerate while you do other stuff, or don’t (I put in a load of laundry, read a bit of math, and started dinner, which took about two hours).

Throw it in your ice cream maker and let it whir for 20 minutes until it looks good.  Put in an airtight container in the freezer.  It’s so good.

## Good Will Hunting “open” math problem (ha)

30 May

I put an edit in italics to show you where I messed up: maybe you’ll see the mistake too before reading the italics!

Fun little post!  I say “open” problem because that’s what they call it in your favorite Ben Affleck-Matt Damon movie*, but the problem they pose is actually super easy and by the end of this post, you’ll be as good at it as Matt Damon!

Here’s the clip from Good Will Hunting:

If you pause the clip at 2:22, you’ll see what the “open problem” is:

Draw all the homeomorphically irreducible trees with n=10.

And that’s the problem we’ll be dealing with today!

So let’s review/learn what all these words mean.  Remember, a graph is a drawing of dots connected by lines (see these two posts for examples).  A graph is connected if you can follow a path from any dot to any other dot.

Left is connected, using the light colored lines (ignoring the three isolated points). The graph on the right is not, since there’s no path from the eye to the mouth.

Now a tree is a connected graph without any loops in it: that is, there’s no way to start heading out from one dot and get back to that same dot without backtracking down the way you came.

Everything but the “NO” is a tree- there are no loops in the graphs.

One funny thing about graphs: it doesn’t matter which way you draw the graph: all that matters is the way the vertices connect with each other.  So for instance, all of the following are the same graph.

No matter how you slice it, it’s still a chain of 9 green dots.

So our GWH problem has to do with drawing a bunch of loop-less graphs, with n=10 dots.  There’s just one last bit of the problem to learn: just like in the topological setting where we learned about it before, a homeomorphism is a function that sends one picture to another picture by wriggling it around continuously.  This is basically what we did in the picture above.

So when I say give me a list of homeomorphically irreducible trees, I want none of the trees you give me to be “reducible” via a homeomorphism.  Rather than a strict definition, I’ll just say that here, homeomorphically irreducible means that none of the dots have exactly two neighbors- if you had such a picture, you could just “erase” that dot and have a picture with about the same amount of information.

The left two graphs are homeomorphically reducible. The rightmost one is irreducible, though that doesn’t seem to make it happy.

Instead of diving in to n=10, let’s try something a little smaller, like n=3.  Here are all the homeomorphically irreducible trees with n=3:

Ha ha it’s a joke!  There aren’t any!  If we want to be connected with three dots, there are actually only two possible configurations: a line (maybe kink it around a bit, but it’s homotopic to a straight one), and a triangle.  The triangle is a loop, so we can’t have that on our list.  The line segment automatically has a vertex in the middle with two neighbors, so it’s not homeomorphically irreducible.  So to answer our question, there are no homeomorphically irreducible trees of degree 3.

Okay let’s try something more interesting.  Here are all four trees for n=5:

You can tell by the way I track that I’m a tree graph, no time to loop back. Ah, ah ah ah, staying with five, staying with five. Ah ah ah ah, staying with fiiiiiiive.

NOTE!  The blue “F” is homotopic to the blue “4″: try to see how you can swing a leg down and reflect for the homotopy.

Here’s the mistake!  There are only three trees for n=5.  The bottom left one is the same as the blue F and blue 4 by straightening out the kink.  Thanks to Dan in the comments for pointing this out.

Which of these is homeomorphically irreducible?  Just the last one: all the others have a vertex with only two neighbors.

If you want, you can draw all the homotopically distinct trees for n=6, 7, 8, 9 in this way and get to n=10, but let’s try to find a shortcut, shall we?

For this next part, we’ll be talking about the degree of a vertex: the number of other vertices attached to it.  For instance, in our bottom right blue “X” graph, we have four dudes with degree 1, and one vertex with degree 4, since the middle one is connected to all four of the outside guys, but each of them only sees her in the center.

So if I go through all the trees with n=5, I have:

1 graph with 2 of degree 1 and 3 of degree 2 (no good)

2 graphs with 3 of degree 1, 1 of degree 2, and 1 of degree 3 (no good)

1 graph with 4 of degree 1 and 1 of degree 4. (ding ding ding!)

What if I go through all the trees with n=4?

1 graph with 2 of degree 1 and 2 of degree 2 (no good)

1 graph with 3 of degree 1 and 1 of degree 3 (ding ding ding!)

This is just a bunch of data as it is, but maybe if I stare at it long enough I’ll be able to find a pattern that will help me with the problem.  I’ll tell you how I did this problem (literally on the back of a piece of paper right now), and send you to a link as to how to do it using equations.

Looking at the data, I notice that it’s a lot easier to start with the highest degree I allow, and build all the graphs from there.  So for instance, with n=5, I can allow highest degrees of 4, 3, and 2 (and allowing 2 is silly because I don’t want vertices of degree 2).  So I only need to look at graphs with degrees 4 (there’s just the star) and 3 (the upper right and bottom left, which both necessarily include vertices of degree 2).

So for our problem, let’s look at the highest degree possible for a graph, and see how many homeomorphically irreducible trees there are with that degree.

Highest degree 10 is impossible, because you need at least 11 vertices for someone to have 10 neighbors.  So we start with 9, which is just a star.  Add it to our list.

How about 8?  Well, if I start with a vertex and give it 8 neighbors, I’ve got one more vertex to add somewhere.  I can’t add it to the central one because it’s all neighbor’d out.  If I add it to one of the 8 neighbors, I get a vertex of degree 2.  So there are no homeomorphically irreducible trees of size n=10 with highest degree = 8.

However, I can make one with 7.  Use the reasoning above, but since we’re trying to avoid vertices of degree 2, I need to add both of my new vertices to one of the 7 neighbors.  Then do the same with 6- I can’t split the three new vertices up between different neighbors, since that’ll also result in a vertex of degree 2.

Here’s a picture of the cases so far.

I bet Kelly Clarkson would be into these graphs: they’re spreading their wings and learning to fly

Not gonna lie, I am getting mighty tired of MS Paint around now.  Phew!  I’ll just tell you the degree sequences for the rest of the graphs, and you draw them, how about that?  That way you’re more like Matt Damon anyway.

For highest degree 5: 5, 3, 3, and seven vertices of degree 1.  5, 5, and eight vertices of degree 1.  5, 3, 3, and seven vertices of degree 1 (there are two graphs with this degree sequence, but they’re different from each other).

There are two graphs with highest degree 4, and two graphs with highest degree 3.

So in total, we have the three trees I drew, and the 3+2+2=7 trees you drew.  That means that we end up with 10 trees, which is exactly what GWH has.  And since we went through all possible combinations of highest degrees and used lots of combinatorial thinking, we’ve proven to ourselves that this is a complete list.

For an answer without proof, see this video from last year (I didn’t know about it when I started this post).

For a more mathy proof, see this link, apparently this answer is due to this Swiss mathematician.

*From googling for the above two links, I realize that in the movie they don’t say this is open, but that it took the MIT math professors two years to solve.  I love it!  Also, hello from Ann Arbor, Michigan!  This is just an explanation for the delay- still traveling, just at a fantastic math conference.