Great Problems for High School

Sometimes I come across problems that make me wish I was teaching High School again.  I mean I don't want to grade papers or go to staff meetings or get up every morning at 5am like I used to; but the idea of watching a bright class of kids thinking about a problem that is just different enough to make them use some of the skills they have other than their great memory was always exciting, and I guess I'll never get over it.

Ever once in awhile I come across a problem that makes me want to pop them on a HS class, but since I'm retired and too lazy to go back to work full time, I thought I would share a couple of the recent ones with the folks still out there working the front lines in case you may have missed them with all the papers to grade and such.

Not too long ago James Tanton reminded me of an old problem about chess boards  by giving a new one I had never seen.  The classic problem I refer to is the one that asks, "Is it possible to cover an 8x8 chess board with dominoes (1x2 rectangles) if the opposite corner squares are cut off.  You, and probably your clever students have already seen this, but it is still a clever problem because of the symmetry idea in the solution. Tanton threw out the question, "In tiling an 8x8 board with dominos, must there be two dominos making a 2x2 square?"
Now at this point I admit I haven't even solved the question, and I'm not sure if Professor Tanton has, (but bet he has). My first instinct is that there must, but I haven't spent the time to test the "Why?" of that. You see, that would spoil it a little. I guess I eventually will, but presenting it to a class when you DON'T know the answer makes the discovery even more exciting. You let the students work without the temptation to rush in and "guide them" to an answer. My experiences in such situations always made me proud of the kinds of thinking my kids could produce when the problem was not "textbook".

Another I saw recently was on Greg Ross' Futility Closet blog. He posted "(5/8)² + 3/8 = (3/8)² + 5/8."

Now giving this to students is not actually a question, unless they have a mathematical mind, in which case they will ask the question; "What does this imply?" For me the immediate question looked like two fractions a/c and b/c so that if you added either to the square of the other the results would be equal. Now the numbers a,b, and c that make that happen would be the question of interest. This might be at a slightly different level than the previous problem, but I keep thinking both would be appropriate from 7th grade to the last year of High School.

The most recent is from John Allen Paulos twitter where he posted a complete proof in the 140 characters allowed. The question was prove that there must be some irrational numbers a and b so that a^b is rational. I'll give you his solution to this one because it is one of a couple that should be exposed to advanced level math students in HS so that they can see some of the beautiful proofs that are out there:

Paulos proof: expanded beyond 140 characters for greater readability, Let a and b both be sqrt2 (irrat.)
Now it may be that c=a^b is rational. If it is, we are done; but if not, then c ^b = 2.
That is not one that will pop out as easy to most students. I played with it and decided that I would try to explain it like this:
sqrt(2) = 2^1/2 so sqrt(2)^sqrt(2) = 2^1/2sqrt(2) and using the product of powers we can get 2^sqrt(2)/2. By using sqrt(2)/2 = 1/sqrt(2) we finally arrive at sqrt(2)^sqrt(2)= 2^1/sqrt(2).

I'm thinking that after this they will be able to see that raising that to the power of sqrt(2) will give a result of 2, a perfectly rational number.

I think at this stage in their lives many of them find rational, irrational, transcendental, imaginary and such a bit mystifying, and some controlled experiments let them gain a little confidence. I'm reminded of a recent blog I read where a teacher/researcher talking to two kids sitting across from each other asked one if she knew a name for the shape in front of her. It was a triangle arraigned so that from her view it was a nabla (∇)(had she been at all aware of the word). She replied that she only knew that from where here classmate was sitting across the table, it would be a triangle. Imagine how many times she must have seen a triangle without seeing one in different orientations. True learning spins on such delicate wheels. You can say the terms as many times as you wish, but when you get your students to describe their views of them, they may start to get a deeper understanding of what they are dealing with.
And if time allows, it is always a wonderfully amazing thing to walk students through the remarkable fact that iⁱ is a real, and infact, is equal to e^-π/2. Do point out that this is real, but not rational.

One disappointment with both of these examples is they really illustrate that there exists a number a of a certain type (irrational for instance) such that a^a has a certain property (rational for instance)  . I think it would be nice to have a collection of irrational pairs a,b such that a^b are rational and demonstrably so at a high school level.  Would love to hear your suggestions?

One final one just because it demonstrates how beautifully geometry can make some math problems visual. This one is also from Greg Ross.
The question is, "How can six people be organized into four committees so that each committee has three members, each person belongs to two committees, and no two committees have more than one person in common?" The question reminds me of Kirkman's schoolgirl problem which involved 15 girls walking in groups.
The geometric solution is easy if you start with the committees as lines in a plane so that no three are concurrent. Then each of the six intersections represents a person and the problem is solved.

OK, Just one more. A short time ago I came across a neat trick by Martin Gardner in Ivars Peterson's column, the Mathematical Tourist that I had never seen, and I thought I had read Gardner's stuff.
 And since all kids love Mobius strips, and this one was even new and surprising to me, I thought I would share.
Start with a simple cross of paper (make it kind of large as some cutting is involved) as shown in the illustration at right.
 

Take one cross and do the usual half-twist to make a Mobius Strip.  The other is just made into a conventional loop with no twist. 
NOW, trisect the Mobius band, and bisect the normal loop.. shake it all out... and be amazed.  Then share it with kids..

Tic Tac Toe.... Looks Like the Number 15...

Playing around with the mathematics of Tic-Tac-Toe with some kids to talk about symmetry and counting... got me thinking..... and searching....

I found the following (slightly edited) interesting history notes at the ADIT sight..

"It may be that the ancient Roman game of Terni Lapilli was an identical game although the evidence is somewhat mixed. It is certainly true that identical grids to the noughts and crosses grid have been found scratched and etched into surfaces all over the ancient Roman empire. However not a single Nought or Cross has been found to confirm the link. It seems probable that Terni Lapilli was played with simple pieces and may have been played with the same rules but in my mind it’s sheer popularity casts doubt upon the connection.

The first software program designed to play Noughts and Crosses (Which is how the English describe T-T-T,... and I recently was told that many Irish folk call the game Boxin' Oxen) was written by A.S. Douglas as part of his PhD dissertation on Human-Computer interaction. The computer was the EDSAC machine built at Cambridge University in 1949. The EDSAC machine was the first true programmable computer as we would understand it today." Here is what the output looked like on the old Edsac..

There are 255,168 possible games of T-T-T, if you label the positions on the board (from 1-9 for instance), but if you allow for symmetry, there are only 138 possible outcomes. If one (or both) of the two players play the game poorly then it may end in a win on the fifth, sixth, seventh, eighth or ninth move....91 wins by the first player (X) and 44 by the second (o)..... and if you are good at arithmetic, then you now know that out of all the games ever played that ended in a draw.... there are only three different final boards. Somehow I think that is incredible. Can you find them all? It is easiest if you just figure out how to place the four " o " so that X can't win...

You can play against an online computer here...
This is a breakdown of all the endings as presented at Wikipedia...
"Ignoring the sequence of Xs and Os, and after eliminating symmetrical outcomes (ie. rotations and/or reflections of other outcomes), there are only 138 unique outcomes. Assuming once again that X makes the first move every time:
91 unique outcomes are won by (X)


* 21 won by (X) after 5 moves

* 58 won by (X) after 7 moves

* 12 won by (X) after 9 moves


* 44 unique outcomes are won by (O)


* 21 won by (O) after 6 moves

* 23 won by (O) after 8 moves


* 3 unique outcomes are drawn"


Now for the 15 part... you can play TTT without the board... Players alternate picking numbers from the integers one to nine inclusive. The first who has picked three numbers that add up to fifteen wins...How is it Tic-Tac-Toe? Well label the TTT board like this....Ohhhh, like MAGIC...

Patterns in Iso-perimetric Problems - 3

In my last post on iso-perimeteric problems I finished with the question "If you have 360 feet of fencing and want to build a pen along a river or barn, the rectangular pen above is NOT the most efficient way to use three straight sections... assume you are limited to four fence posts, so you put two somewhere on the river, and can put two more along the fencing to hold straight sections of fence; how do you place them?

Now extend that problem to four sections of fence (and five fence posts)... and then generalize the results... "

The sweet mathematical idea to make all these types of problems easy is symmetry . Imagine that you had found the perfect solution to the fence problem with three sections and used it on your side of the river. Then the guy on the other side of the river decided to copy your master plan on his side...
NOW make the river really narrow..... no..more narrow... like a line.. Now erase the river completely...
What do you see when you look at your neighbors fence and your fence together? It must be a six sided figure.. and it must contain the most area it could contain for any six sided figure, since both sides hold the maximum possible... you couldn't make the total bigger without making one side or the other bigger...so you must have the most efficient six sided figure possible, a regular hexagon.
So what does your half look like? It must be half of a regular hexagon divided across two opposite vertices.
The solution would be to divide the 360 feet of fencing into three sections, and make the angles between the fence sections 120^o and slide the two ends up against the river.
To solve the three sided non-symmetric figure, we used 1/2 of a six sided symmetric figure. And the general solution from there, I hope, is trivial.

Actually with a little handwaving, this blog could have been much shorter..but I remember Pascal once wrote, "I have made this longer than usual, only because I have not had the time to make it shorter." [Lettres Provinciales, No. 16 (1657)]

Close order Drill and the square root of -1

I’ve been thinking about groups lately, both the mathematical kind and the social kind. The thoughts were prompted by another trip with some of my kids to the Center for Mathematical Sciences in Cambridge to hear Professor Marcus de Sautoy speak on his new book, Finding Moonshine, which is soon to be released [the American title seems to be different] and is about symmetry and mathematical group theory. When mathematicians talk about symmetry, we talk about groups.

The social group that joins me on my trips seems diverse at first glance. There are a few seniors, a few juniors, and a smattering of sophomores. There are about the same number of boys and girls, and they are as different as high school boys and girls can be. The girls will giggle at themselves, amazed that they can be excited about going to a math lecture, while the boys grow more quiet than usual when they are out of their natural turf. We stand outside one of the lecture rooms and study a board covered with symbols that neither they nor I can decipher. They try to recognize some of the symbols, a “!” factorial symbol here, “and” and “or” logic symbols there. “Is that Sigma for summation?” Some one suggests its some kind of probability problem. “What’s that one, With the Pi in the parenthesis?” Grateful that I actually know one they don’t, I explain, “That’s a capital Gamma, a function like factorial that works for all real values, not just integers.” They say “Ahh” as if they understand. We’ll expand on that on another trip perhaps, a day when an expert can peal back another level of the mathematical mystery just a bit.

What makes some kids, some people, open their minds to the complexities of math, drawn to the cryptic symbols they don’t understand? It must be the same drive that led Champollion to decipher the Hieroglyphs. How is it that one kid can be blinded to the relationship of imaginary numbers by the simple hurdle of its name, while another wants to visualize a snowflake that exists in a universe with 196,883 dimensions.

All these thoughts wandered through my mind as I watched a group of ROTC students passing beneath my window and stop at attention in the open courtyard below. As they practices the simple stationary turns common to such formations, “A’ Ten Hut”, “Right Face”, About Face”, “Left Face”, I realized that I was watching them perform a physical demonstration of the same relations they would swear were too complex to understand in their algebra classes.

The mathematical term Isomorphism is from the Greek roots for “same body”. The parade ground moves represented an order four group that was isomorphic to the multiplicative relationships between the imaginary quantities they found so impossible to comprehend. The four activities on the drill pad could each be paired with the four primitive mathematical quantities, 1, -1, I, and –I, so that they each would produce the same result.

One is the identity, Like “Attention” it keeps the position fixed. About face is the same as -1. If we think of right face as i, the square root of -1, then left face would be –i, the opposite of i. Any two actions on the pad operated like multiplication of its counterparts in the abstract number set. About face followed by about face was like Attention, not turning at all, just as multiplying -1 by -1 returned us to the mathematical identity. Right face followed by about face produced left face, and mathematically -1 x I gives –i. What about that mysterious i x i that so confused them in the algebra class. Right face followed by right face was just about face, the drill pad’s symbol for -1. No student in the small squad in front of me would have thought it difficult to imagine what his position would be if I asked him how he would be standing if he made back to back right faces.

The same abstractions that made math so powerful, that allowed us to represent the drill team, and the multiplication of complex numbers with the same symbols was a beauty they could not see. But some will twist their faces and squint against the dimensional curse of being born in three-space, hoping to get a glimpse of a four-space hypercube, one step closer to the Monster. Only 196,879 more dimensions to climb. But on the way home, they start with baby steps. Someone asks, “Can there be a group with just two elements?’ I was going to answer, when one conjectures, “Yeah, I think just one and negative one would be a group under multiplication.” They discuss it for awhile, ignoring the old man driving the car; after all, they learned tonight that “a mathematician seldom produces great work over the age of forty”, and if they don’t know exactly how old I am, they know that my age is much greater than forty; or as they would say, my age = .