Tag Archives: math

Calvin! Hobbes! Scrabble!

The first thing to note about this is that it’s impossible.

The total point value of the tiles in a Scrabble set is 187. For the maximum possible score when making this play, assume that he completed six words of fifteen letters. Then he’s getting points from ninety tiles, for a maximum of 177 points before  multipliers.

For a score of 957, the average tile would have to be counted 5.4 times, meaning about 54 points for a Z or a Q, or 5 or 6 for a vowel. The specific six letters he placed  are all doubled (from the double word score) and used twice (because they’re each in a different word as well). So each of those letters counts at least three times. The specific letter on the double word score box counts four times, and if one of the Z,Q,F,M,G,B is on a double or triple letter score then it counts six or nine times. But there are no other bonuses. So the maximum mathematically possible score from placing six consecutive letters one of which is on a double word score is 9*(10) + 4*(10) + 3*(8+8+5+5) + 1*(5+4+… you know what, this is way less than 957.

Far be it from me to suggest that Calvin was cheating. They must have been using a different ruleset. There’s a commonly used house rule that if you lengthen a word, you get any bonuses from any letter in that word. (As opposed to just the bonuses underneath the letters you played.) This rule is wrong and messes up the game, but a lot of people play with it and it does allow for Calvin’s score.

Here’s a possible board configuration immediately before his turn. (You might want to refer at a blank board to see where the bonus squares are.)

If it helps, the first word played was "Leo."

The words played on Calvin’s turn are:

Antibionicities: You know how, if you have multiple bionic limbs you might say you have more than one bionicity? This is the opposite of that. Also, three triple word scores. (1+1+1+2+3+1+1+1+1+3+1+2+1+1+1)*3*3*3=567 points.

Qasidah: Before the Q, this was a food. With the Q, it’s a type of lyric poetry. (10+1+1+1+2+1+4)*2*2=80 points.

Unflakelikely: In a manner not resembling a flake. (1+1+4+1+1+10+1+1+1+5+1+1+8)*2*2=144 points.

Nonglistening: Only capable of hearing things that don’t include the letter G. Wait, that was something else. (1+1+1+2+3+1+1+1+3+1+1+1+2)*2*2=76 points.

Tom: The cat, not the name. Neither Calvin nor I would ever stoop so low as to use a proper noun. (1+1+3)*2 = 10 points.

Pize: This word is really cool. The actual definition is just “a pox or a pest,” but it was primarily used as a curse word. Kind of like “A pox on Insert Thing Here,” but more obscure and archaic. Oh, and as a verb it means to strike someone. 3+0+10+3=16 points.

Zqfmgb: A worm found in New Guinea. (10+10+4+3+2+3)*2=64 points.

(And yes elemental symbols count. And “Ec” counts just as much as com or org, thank you very much. And “nn” counts because Unix programs aren’t proper nouns. And so on and so forth.)

64+567+80+144+76+16+10=957 points.

Hobbes apparently played a word with “all the Xes and Js,” which is weird because a Scrabble set only contains one of each. From this I conclude that they played with tiles from more than just the one set. That happened to my family’s set years ago, so it’s not like that’s impossible. Also there is no X or J on this board outside of the word for female former judges. And I did hold myself to the regulation number of blanks.

(Also: nucleoplasm, be, zygomorphic. It should be possible that they were all played for the right number of points given the rules as I stated them.)

It is very likely that I made at least one mistake. If anyone finds one, I’ll make the correction and sing the Very Sorry Song in your honor. And by “sing” I mean “post, over a text-only medium.”

You Know More Things Than Laplace’s Demon

So, it turns out the average human knows more facts than some varieties of omniscient being.

Picture the least omniscient thing that still deserves that adjective. I’m thinking of something a couple notches up from that.
The thing I’m thinking of is a hypothetical something that knows everything there is to know about the universe and all the objects in it, from quarks to galaxies and beyond. And it knows nothing else. Obviously this thing can fairly be called omniscient. But it doesn’t necessarily have to be intelligent. It might be, or it might have no concept at all of logic or reasoning.

The thing I’m talking about is basically identical to (the weakest possible version of) Laplace’s Demon. So I’ll call it that. It knows everything knowable about the state of the universe and the laws of physics. It can extrapolate backward to see exactly how Socrates died, so you bet it knows he was mortal. And it knows exactly how many humans die. But it might not understand the “therefore” in “all men are mortal, Socrates is a man, therefore Socrates is mortal.”

It can check throughout all of time and space and make sure that every time there are two of something, and two more of that thing, there are four of it. But it doesn’t know that 2+2=4, because it doesn’t understand “equals.” It doesn’t need to. (If it can think, that won’t hurt it. The one in the Trivia Contest could. But it doesn’t need to think to be a functioning Laplace’s Demon.)

I claim that this version of Laplace’s Demon knows at most a countable infinite number of things.
In our universe, it doesn’t make much sense to divide distances into smaller and smaller pieces forever. It’s conceivable that some universe might work that way, but ours doesn’t. In ours, you hit a limit, somewhere around one decillionth of a centimeter. Anything smaller than that limit is literally unmeasurable, as Laplace’s demon knows better than anyone. Two locations separated by less distance than that might as well be the same place, and that is an actual fact about the universe not about measurement technology. (Current tech is nowhere near that good anyway.) And in any finite distance, there is only a finite number of Planck lengths.

Same thing with time. The universe started a long time ago, and the shortest measurable length of time is very small. There have been an awful lot of Planck time intervals since the universe started, and there will be an awful lot more before it ends. But it’s still a finite number.

I, like most currently existing humans, don’t know whether or not the universe is infinite in size. (Laplace’s Demon does.) If so, that’s a countable infinity. It’ll be infinite in the sense of “go one Planck length at a time, keep going forever. For any given point, you’ll be at that point eventually, but you’ll never run out of space.” This is the smallest possible infinity.
It is a property of infinity that infinity times itself is the same size of infinity*, so if the universe goes on forever in all three dimensions and we have to cube that, then we’re still talking about the smallest possible infinity.

Laplace’s Demon knows everything about every point in the universe at every time. It can look at any space-time coordinates and tell you exactly what particle, if any, was/will be there. (Can there be multiple particles within the Planck distance of each other? I should probably know that, but I don’t. If so, the Demon can tell you exactly which ones were where. Anyway, I’d be surprised if there can be an infinite number of particles in the same Planck volume and shocked if it can be an uncountable infinity.)

We’re multiplying a countable infinity (from the size of the universe, measured in Planck units) by large finite numbers (from the duration of the universe, and the number and type of particles in each space). So the result is still a countable infinity. And if the universe turns out to not go on forever, then Laplace’s Demon only knows a finite number of things. That’s almost impressively small.

By comparison, you (yes, you) know an uncountably infinite number of true things. Watch this: “For any real number greater than three, that number is greater than two.” That single obvious sentence generates an infinite number of things you know. You know that 4>2, that 3.001>2, that π>2, and so on. There are an infinite number of things you know, and that infinity, the uncountable number of real numbers above three, is bigger than the countable infinity of things that the Demon knew. Since humans have the awesome and underrated power of abstract thought, we don’t need to limit our knowledge to concrete facts.

In fact (and you can skip this part if you want to avoid math vocabulary), I’m pretty sure that the set of true statements that I know has cardinality Beth Omega. That’s big, even for an infinity.
Let Xn be the set of maps from Xn-1 to Xn-1. X0 is the natural numbers. The number of maps is the cardinality of Xn.  Since each Xn is the powerset of the one before, the cardinality of Xn is  \bethn.

\beth is the Hebrew letter Beth; \beth0 is the cardinality of the natural numbers. n can be any finite natural number.  \bethn means 2בn-1. (That’s two to the power of the previous infinity.) The smallest infinity larger than all \bethn is called \beth_\omega. Lower case omegas don’t look nearly as cool as the capital ones. For any n,  \bethn+1 is bigger than  \bethn. So the cardinality of the union of Xn over all n must be greater than any \bethn. It is therefore at least \beth_\omega

I happen to know that every map in that infinite union is the image of another one. (Trivially true: For an arbitrary map in Xn, there is a map in Xn+1 that just turns all maps in Xn into that map. So any given map in Xn (for any n) is the image of something.) Therefore, there are at least Beth Omega true statements I can claim to know.**

Even without the confusing part, I think it’s safe to say that a normal human knows an uncountably infinitely larger amount of true things than that omniscient Demon did. (No comment on who gets a better selection of things.)
So; next time you want to win an Internet argument by appealing to credibility, make this claim. Tell people your knowledge exceeds that of some omniscient beings, and that it would be impossible to list all the things you know even if you had infinite time to do it inYou’re welcome.

*Sort of. The cardinality of an infinite set crossed with itself is equal to the cardinality of that set, but the usual kind of multiplication doesn’t really make sense with infinities. But if you promise not to repeat this in front of any mathematicians and definitely promise not to tell people that said it, then sure. Infinity times itself is the same size of infinity.

**The previous version of this claimed that the number of continuous functions from Xn to Xn is \bethn, which is a smaller infinity than \bethn+1. It feels like that’s probably true for some form of continuity, but I didn’t think it through. If the old version actually is right, I got lucky. Would this be a good time to mention that the number of things I don’t know is also hugely infinite?

I bet he broke the sound barrier

(Spoilers for the Dark Knight Rises.)

At the end of the movie, Batman and his allies fail to stop the nuclear bomb, so he flies it out over the bay to prevent it from wiping out Gotham. People on the Internet correctly noted that this is definitely underestimating the range at which nuclear weapons can cause Bad Stuff to happen. But there definitely exists some distance such that the bomb going off at that range wouldn’t hurt anyone in Gotham to any instantly noticeable degree.

At the football scene, Dr. Scientist states that it’s a neutron bomb with a blast radius of six miles. Bane said it was a four-megaton bomb, and those numbers do not go well together. I’ll take the nuclear physicist’s word over Bane’s, and we can find out how far Batman must have gone.

I don’t fully understand why the equations I’m using are the right ones (I would have expected inverse squares instead of weird fractional exponents), but Wikipedia cited this as a source so it probably works. Anyway, for a neutron bomb (unlike most nuclear weapons) half the energy goes to radiation and does not affect the size of the blast radius. If the blast radius (in km) is (Yield/2.5kt)^.33, then a six-mile blast radius means 2410 kilotons. Double that, because half the energy is going into radiation instead of blast, and it’d take a 4.82 megaton bomb. So Bane wasn’t that far off after all.

We can determine how far Batman would have to fly it. It was obviously over six miles, or everyone watching it would have been blasted backward and very possibly killed just from the air. But it must have been even farther than that, because there were unprotected humans with a clear line of sight to the blast. When it went off, they cheered. They didn’t appear to have been covered in burns or struck blind or anything.

For a 4.8 Mt neutron bomb, the thermal radiation would hand out third degree burns within about a ten-mile radius. But the other effects of the thermal radiation reach go further than that.

The people on the bridge were all watching Batman fly the bomb away. And they were looking directly at it when it went off. (Note: Never do that.) Fortunately for them, it went off during the day instead of at night. Dilated pupils would be a bad thing. Unfortunately for them, it was a pretty clear day. A one-megaton nuke would temporarily blind people from 13 miles away. That’s for a regular nuclear weapon where 5% of the energy goes to radiation. For a neutron bomb, that number is 50%, leaving proportionately less for thermal radiation. (To make up for that, the neutron radiation is worse, but that has a smaller range anyway.)

Since a two-megaton neutron bomb would blind people from a bit over 13 miles, a 4.82-megaton one would blind people from 13√(4.82/2) miles away.  That’s a bit over 20 miles. Hopefully he flew it farther than that, but any closer and the kids in the school bus would definitely not have been cheering. So this gives us a good lower bound on his speed.

The timer on the bomb showed 1:57 when Batman attached it to his flying car. Then he kissed Catwoman and told Commissioner Gordon his not-remotely-secret-anymore identity and started the car, and by the time he took off it had been over 40 seconds. That leaves less than 77 seconds for him to fly it more than 20 miles. Apparently the Bat can fly at 935 miles an hour, which is well over the speed of sound. So I win my bet, muahaha.

Maybe there are other effects with wider reaches. Like, Gotham is probably going to have some severe fallout problems later. If there are effects that would show up on screen as soon as the bomb goes off and have a wider range than the flash blindness, Batman would have had to take the bomb even farther. But 20 miles in 77 seconds gives us a lower bound: He must have gone at least that fast.

(And it just occurred to me that I should be timing from when he passed the bridge instead of when he took off, since the bridge is where the people were. But the movie didn’t show the timer position for that, so I’ll just say it’s definitely significantly higher than 935 and use that number anyway.)

Incidentally, if he covered that much distance in that little time, then his average acceleration was 20/77*3600/77 miles per hour per second. Or 5.4 m/s^2, about half a gee. Which is much more survivable than I was expecting.
But the air resistance against the giant spherical bomb would be .5*1.225 kg/m^3*(418 m/s)^2*.47*π*.75^2 = 88.885 kiloNewtons. That’s about ten tons of force just from the air pushing back against the bomb. And the hovercraft can apparently fly at over 935 miles an hour while dragging that behind it. Too bad it got nuked, because that must be a seriously awesome machine.

Nerd Snipe

The post I was planning on putting up today got delayed by completely foreseeable events, so here’s a cool nerd snipe I ran across recently.

Start with a shape that looks like a right triangle except that the hypotenuse has been replaced with part of the arc of a circle. (Convex inward, and intersects the triangle at only those two points.)

Screen shot 2013-12-04 at 9.59.03 PM

Like this but less blurry.

Prove that the curve is shorter than the sum of the two sides.

I like this question because it’s immediately obvious just from looking at it, but you’ll probably have to think to get a proof. It seems like there should be a completely obvious proof that you could explain to a six-year-old, and there probably is, but if so I don’t know what. Even the best proof I’ve seen involved a limit and wasn’t all that satisfying. Can you find a good one?

(As incentive for trying, a proof here will also work as an explanation of the troll pi phenomenon. Because there’s only a countably infinite number of points where the shape with all the right angles intersects the circles, and there’s an uncountably infinite number of points on the circle, there must be some points where no matter how many iterations you go, that point will never be on one of the corners. So zoom in on one of those points, and you get this shape. Proving this will prove that the shape with the right angles has a larger perimeter than the circle, and Archimedes will praise you as the savior of mathematics.)

Anyway, the post that was meant for today will probably go up in a day or two.

Fencing to Chess Rating Comparison

Last weekend, I was at  a chess tournament and a fencing tournament. At the chess tournament, I could tell approximately how good someone is by the number after their name, but had no frame of reference for the fencing rating system.

At the stabbing competitions, some of the competitors have a rating, a letter after their name from E to A.  The problem is, it doesn’t really mean anything to anyone who doesn’t already know about it. If someone has a B rating, does that mean they’re tough but beatable, easy, or you-should-run-the-other-way-and-scream?

I’m familiar with this problem from failing to explain what a particular number in a chess rating means, but I’m not used to being on this side of it. I had no idea how impressed to be at any point. So, in the “things that will probably interest precisely nobody else” category,  I decided to try to come up with some way to convert between the USFA and USCF ratings, just so it makes sense to me. I have no idea if I succeeded.

Some cursory Googling turned up this forum post from the distant past, saying that about 2% of epee fencers have an A rating. (1.99% was the biggest number available. I should probably be more unsettled than I am by the contradictory numbers citing a different now-unavailable source, but I decided to use the bigger one and call it a margin of error.) The vast majority of fencers are unrated. In chess a rating is a prerequisite for all the good tournaments; in fencing you earn one.

The chess people have some conveniently available data, and from an official source, no less. (Oh yeah, these are just American organizations, if the “US”-es in the acronyms didn’t give that away.) This is out of date by about the same amount as the numbers I’m using for the fencing part.

Before comparing the numbers, I’ll get all the “I don’t know what I’m talking about” disclaimers out of the way first, and then go on to talk as if I’m definitely right. First: I don’t know what I’m talking about. I’m basically completely unfamiliar with the USFA rating system. Second: I’ll be pretty much assuming that whoever has the higher rating is better at their sport. This is only true on average. And I’ve heard that upsets are more common in fencing than in most sports, especially if you’re fencing epee. Third: If I told you that skill at fencing and skill at chess are as different as apples and oranges, you’d laugh in the face of my understatement. All I’m doing is comparing percentiles. Third and a Halfth: You can compare percentiles of anything. I’m using chess and fencing ratings, but it could just as easily be marathon time and number of blinks per minute. There’s no actually meaningful quantity being measured. Fourth: I don’t have access to the original source for the fencing numbers, so I hope nobody minds hearsay.

So, those are the reasons I don’t actually trust these calculations. With that out of the way, I am now completely right about everything and definitely have all the numbers I could possibly want.

I want to compare how good someone is at fencing to how good a chess player is, because the chess ratings are numbers that actually mean something to me. So, compare the percentiles. An A rating for an epee fencer means they’re in the top 2%. (1.99, but who’s counting.) For a chess player (use the non-scholastic column for more fair numbers), the top 2% would mean a rating a bit over 2200. So by my calculations, losing to an A-rated fencer is approximately as embarrassing as losing to a low-level chess master. (For most people, this means, “not very.”)

Now assume that all rated fencers are uniformly better than all unrated fencers. Then a B rating for a fencer corresponds to the top 5.15%, which is about the same percent as a USCF rating of 2000. That’s just about exactly expert level. Do the same thing with the other fencing ratings, and you get that C corresponds to 1900 level, what chess players call class A. (Just to be confusing.) And fencing rating D means a slightly lower-rated class A chess player, and E to a low class B. This means that there are several categories full of people who are better at fencing than I am at chess. My ego will not stand for this; fortunately it doesn’t have to.

In real life, of course, not all rated fencers are better than all unrated ones. At high levels it’s true, because anyone who can compete that well is probably experienced enough to have earned a rating. But at lower levels, it’s entirely plausible for someone to be good enough to have a rating but not have it yet. Let’s assume that instead of all unrated fencers being worse than all rated ones, they’re actually just as skilled with the same distribution. So if 10% of rated epee fencers have an A rating, we’ll take this to mean that 10% of all the competitors are that skill level.

Now an A-rated fencer is only in the 90th percentile, corresponding to not a chess master nor even an expert but somewhere in the middle of chess class A. For everything else, a fencer with a rating in some letter would be in a worse percentile than a chess player with the similar-sounding title. An E-rated fencer would be in the same percentile as a chess player rated 800. I’m not even sure that someone who played at that strength could beat non-chess-players reliably, and an E-rated fencer has done well in at least one fencing tournament full of fencers who fence.

The most accurate comparison will be somewhere in between the two, pretty heavily weighted toward the first estimate. (The one that’s good for fencers.) It’ll get even closer to the first estimate as the ratings get higher. A fencing rating of “A” probably does correspond more or less to chess master level, so I don’t feel too bad about losing that one bout five to zero in seventeen seconds.

Drops of Jupiter

I heard a song the other day. Half the lyrics sounded like the sort of meaningless but positive-sounding things you expect out of song lyrics, and the other half were astronomy. Bad astronomy. I utterly failed to take the song seriously, because whoever it’s being sung to is very extremely dead.

“She’s back in the atmosphere with drops of Jupiter in her hair.”
Believe it or not, saying “drops of Jupiter” actually makes sense. There exists a substance that, while not unique to Jupiter, is characteristic of it. It makes up a large fraction of the planet and is in fact liquid. Perfect. That material is metallic hydrogen. So naturally the question is, what happens if a drop of metallic hydrogen from Jupiter were to appear in someone’s hair?

Not pictured: Safety.

At the shallowest relevant depth in Jupiter’s atmosphere (there are much bigger numbers available, but we want to give the lyric-writers the benefit of the doubt), the temperature is 10,000 Kelvin, and the pressure is hundreds of thousands of gigaPascals, millions of times Earth’s atmospheric pressure. We’re talking center-of-the-earth type pressure and temperature hotter than the surface of the Sun. It should go without saying that you do not want that in your hair.

Wolfram|Alpha tells me that a “metric drop” is a unit of volume. I needed a number, so I may as well use the actual one. So this person has a couple of drops of metallic hydrogen in her hair, each drop being .05 cm^3. To find the actual amount of hydrogen in each drop, PV=nRT. Yes, that’s an equation for gases not liquids, but this temperature and pressure is a transition point where it’s going between those phases, so I’m hoping it’s still valid to use it here. (Despite the term “metallic,” this is kind of like a gas and kind of like a liquid and not remotely solid.)

If this equation doesn’t work here because the material is too liquid, well, the liquid phase is obviously denser than the gas anyway, so this will be an underestimate rather than an overestimate. So just pretend I added another clause on to each paragraph saying that it’s at least this bad.

PV=nRT. (2*10^8 kPa)*(.05 cm^3)=n*(8.31×103 cm3*kPa*K-1*mol−1)*(10000 K). Solve for n, and it’s 0.12 moles of hydrogen. Thanks to hydrogen’s excessively convenient molar mass, that’s also 0.12 grams. That…doesn’t sound so bad. Until you remember the temperature and pressure it’s at.

The first thing that happens is that this small drop immediately and explosively expands until it’s at the same pressure as the air around it. Since the pressure is dropping by a factor of 2*10^8 kPa/101.3 kPa =1974000, the volume increases by the same factor. So that’s how many metric drops of volume it occupies, and that means it takes up about three and a half cubic feet. Not a huge amount, but enough that when the explosion happens right next to someone’s head, well, it does not go well for them. I might run those numbers later.

The next thing that happens is that the hydrogen sublimates and cools down. It goes from being in its metallic liquid phase to being diatomic gaseous, and then since it’s surrounded by air that’s cooler than it is, its temperature decreases. Hydrogen has a specific heat of 14.3 J/g/K, meaning it takes 14.3 Joules of energy to raise the temperature of one gram of hydrogen by one degree Kelvin. Conversely, for each degree Kelvin that the temperature decreases, there are 14.3 Joules per gram of hydrogen released. That energy has to go somewhere.

The specific heat is actually higher, because these are some pretty extreme temperatures and it’s not really a constant. But I’ll round down so the numbers are simpler, and add a third “it’s actually worse than this” disclaimer. We’ve got 0.12 grams here, so each degree of temperature change sends 1.716 Joules of energy into the air or whatever else is unfortunate enough to be close.

Since the temperature starts at a staggeringly high 10000 K, and is going down to regular Earth atmosphere temperatures like 300 K (80º F), that’s a change of 9700 K. So it’s releasing more than 16.6 kJ of energy in the form of heat to whatever happens to be nearby. That’s surprisingly small; Wikipedia tells me it’s about the energy contained in nine M16 rifle cartridges or four grams of TNT, but to be honest I was expecting something more apocalyptic. But it does happen to be centered on a human head, which is bad.

A human brain masses about 1400 g, and has a specific heat of 3.64, so if all the heat were going in to the brain and were evenly distributed, it would only raise the brain temperature a few degrees. Which, sure, could kill the person. And yes, it won’t be evenly distributed. The heat will be concentrated at the skin near the drop. All the skin on that side of the head will be instantly destroyed and I don’t know what will happen to the skull. But it’s a lot less bad than it could have been; I was expecting the heat to leave her head completely vaporized.

Well, that was less bad than expected. Let’s stop worrying about the heat and look back at the force of the explosion.
The center of the blast, of course, was close enough to the person’s head to be described as “in her hair,” so a significant fraction of the incredibly hot exploding gas is going to come into contact with her head. It’ll be less than half, but in about that ballpark.

A sphere with a volume of one drop has surface area 6.56*10^-5 square meters. The pressure in that drop was 2*10^8 kPa= two hundred billion Newtons per square meter. So the force being directed headward is about half of the product of those. (Can I use these numbers this way? It seems like I should be able to.) If that much force hits the head of the person in question, that could be millions of Newtons. Like, the kind of force like when you’re minding your own business and suddenly there’s a blue whale on top of your head. That kind of force. It’s even worse than I was expecting.

To make things even worse, the singer said drops. Plural. It is very safe to say that the subject of this song no longer has a head, and the rest of her body is probably obliterated as well. So, as a general rule of thumb, you should probably keep drops of Jupiter away from your hair.

Fortunately Not a True Story

Once upon a midnight dreary, while I pondered weak and weary
Over many a lengthy text of theorems whose proofs to find.
Drawing useful lemmas from thence, by the problem’s antecedents
Showed I then by abstract nonsense that the converse was implied.
“’Tis unfortunate,” I muttered “it’s the converse that’s implied”
As the proof escaped my mind.

Dealing with this foul equation, foiled by obscure notation
Still I saw no sound causation from the left to right hand side.
With this cursed reverse inclusion all I’d need to do was use one
Of the claims I hadn’t proven; they could prove the ones I had.
But unfairly used conclusions that can prove the left hand side—
“Tis it’s converse that I need.

‘Tis but vaguely I remember, it was sometime past September
And by all my class’s members every method had been tried.
Staring at the sheet unblinking, feeling all my chances shrinking,
Near despairing, always thinking “Can I prove the other side?
Is the statement even true that sat upon the right hand side?”
For it still my skills defied.

Midnight passed as time grew longer, certainty grew ever stronger
That the proof was wrong and wronger, for no true proof could I find.
Each as failed as its successor, my attempts were ever lesser
“I must ask the class’ professor what theorem may be applied.
“On the morrow I shall ask him if with success this may be tried.
For my brain’s so liquefied.

“Problem!” said I “Unsolvable! Problem which might be possible!
If solution may be found or count’rexample be espied!
I was up til seven a.m. with all manner of mental mayhem.
By every tactic and stratagem just the converse was implied.
Is this possible, professor? Was a false proof there assigned?”
Quoth the teacher then, “I lied.”