Wednesday, October 11, 2017

PSAT Test (Day 40)

This is what Theoni Pappas writes on page 284 of her Magic of Mathematics:

"The game of Hex was invented in 1942 by Piet Hein. The game board for Hex is made up of hexagons."

This is the last page in the section "The Games Mathematicians Play." This final game, "The Game of Hex," is another highly visual game, so let's provide another link:

Hex is played on a diamond-shaped board made up of hexagons. The game is usually played on a boards of size 11 on a side, for a total of 121 hexagons, as illustrated above. In the game, one player plays white pieces, while the other plays black, with play alternating between players and placement only allowed on unoccupied hexagons. Alternate sides of the board are designated white and black as shown above, and the goal of the game is to complete a chain of pieces between one player's two sides. The game cannot end in a draw since no chain can be completely blocked except by a complete chain of the opposite color.

Pappas mentions an extra rule that's not given at the Wolfram link above:

"As illustrated, the first player is not allowed to play his or her piece in any of the gray hexagons."

This rule applies to the opening move only. There are no gray squares on the Wolfram Hex board, and so this rule doesn't apply to that version. On the other hand, Wolfram actually labels two of the sides "white" and two other sides "black," indicating that a player's winning chain must connect two sides of the indicated color. Pappas, meanwhile, writes only:

"The first player to make a path to the opposite side is the winner."

Pappas provides three pictures of Hex games in various states of progress, each with a caption. There is no point in my typing those captions here on the blog, since you can't see them. So instead, let me write more about the use of Hex in the classroom.

Unlike Rithmomachia, Hex is easily playable in a math class. Indeed, the first time I ever saw Hex played was as a young Algebra I student. An 11 * 11 game board (called a "diamond" by Wolfram, but more accurately a rhombus) is easily drawn and photocopied. In fact, here's a link to a Hex board that I found via a Google search:

Whereas Rithmomachia requires pieces, Hex can be played using X's and O's, like Tic-Tac-Toe. We see that according to Wolfram, Hex can't end in a "Cats Game" or draw.

The old TV game show Blockbusters (which I watched on the BUZZR channel) is based on Hex. It's played on a 5 * 4 board, and just as in the Wolfram version, opposite sides are labeled. In this version there are actually three players -- one "solo player" vs. two, a "family pair." The family pair must play in the longer direction while the solo player goes in the shorter direction.

According to Wolfram, if the shorter player goes second, that player is guaranteed a win simply by using reflections. But the solo player didn't always win on Blockbusters because players must earn their hexagons by answering questions. It's the same reason that Tic-Tac-Toe almost always ends in a draw, but Hollywood Squares almost never does -- players must answer questions to earn squares.

(Oh, and speaking of the BUZZR channel, just today there was an old episode of I've Got a Secret in which the panelists had to solve riddles. One riddle, which stumped the entire panel, was basically the "hiccup" riddle mentioned on the blog two weeks ago, from Pappas page 271.)

I also think that I subbed once in a classroom where the students were playing Hex -- but I can't verify it since I made no mention of it here on the blog. I also thought I once saw some MTBoS math teachers mention Hex on their blogs. But a Google search for MTBoS Hex posts reveals only references to old Twitter posts. As I found out last week when I tried to link to a review activity, Tweets are ephemeral links rather than permalinks, so it's tough to link to them here.

Before we leave Hex, here's a link to another favorite math website, Cut the Knot:

According to this link, it was the famous puzzler Martin Gardner who first made Hex popular. At this link, there is an indirect proof that not only can Hex never end in a draw, but that there exists a winning strategy for the first player:

The argument goes as follows. Assume there is a winning strategy for the second player. We are then going to show that the first player can utilize this same strategy to win the game. To make this work, the first player makes a very first move and, in a sense, forgets about it. He imagines now being a second player and, as such, playing the winning strategy on the board with an extra chip of his color. Note that the extra chip can do no harm to this player. If, in time, the winning strategy requires placing a chip in a cell occupied by the extra chip, the player puts a chip randomly elsewhere, and thinks of the latter as the (new) extra chip. Impersonating his rival and playing the latter's winning strategy will lead the first player to victory. Contradiction.

This most likely explains the "gray hexagons" in the Pappas explanation -- their purpose is to neutralize the first player's advantage. But as often happens so much in mathematics, this is merely an existence proof of a Player 1 advantage -- it doesn't tell us what it is. According to Cut the Knot, winning strategies are known for boards of size 9 * 9 or smaller, but not the standard 11 * 11. The proof was first discovered by John Nash, credited as a co-inventor of the game. (This mathematician is featured in the movie A Beautiful Mind.)

I also found an entire website devoted to the Game of Hex:

And now let's go from a "beautiful mind" to "the man who loved only numbers," Paul Erdos. It's time for us to start our new side-along reading book. I've created a new blog label for Erdos. Like several mathematical books, this book contains a Chapter 0 and several other interesting chapter numbers.

Chapter 0 of Paul Hoffman's Man Who Loved Only Numbers, "The Two-and-a-Half Billion-Year-Old Man," begins as follows:

"Vegre nem butulok tovabb" (Finally I am becoming stupider no more)
-- the epitaph Paul Erdos wrote for himself.

The book opens at a New Jersey restaurant, and Paul Erdos, the famous mathematician, is sitting alone at a table. He is supposed to be sitting with his four colleges, but they are instead sitting at another table. Erdos doesn't realize that they've moved, because he's too busy trying to solve a math problem on his place mat!

This, of course, explains the title of the book. During his entire life, Erdos lives and breathes math, and spends little time doing anything else.

Hoffman, the author, tells us that the mathematician's last name is pronounced "air-dish." This is emphasized in a limerick written about him:

A conjecture deep and profound
Is whether a circle is round.
In a paper of Erdos
Written in Kurdish
A counterexample is found.

He also quotes Erdos when he famously tells us, "A mathematician is a machine for turning coffee into theorems." And when others advise him to slow down, Erdos essentially says that he'll sleep when he's dead.

Erdos apparently has his own language. When he first meets the mathematical essayist Martin Gardner (yes, the one who popularized Hex), Erdos asks him "When did you arrive?" Apparently, this means, "When were you born?" To be "dead," to Erdos, means to stop doing math. He referred to someone who actually died as having "left."

Actually, Erdos isn't Kurdish -- he's Hungarian. And just a few years after his birth (er, "arrival") in the capital city, he is already doing math:

"Erdos was a mathematical prodigy. At three he could multiply three-digit numbers in his head, and at four he discovered negative numbers."

Once again, I can't help but compare Erdos to our drens, who can't even multiply one-digit numbers in their heads. And of course, I'm referring to teenagers, whose math skills are nothing compared to those of Erdos when he is still a young "epsilon." (That's what Erdos calls young children. I explained in previous posts that "epsilon" refers to a small number -- most recently when I was explaining why a random process created fractals.)

Later on, the Hoffman explains the Chapter 0 title as he quotes the mathematician:

"When I was a child, the Earth was said to be two billion years old. Now scientists say it's four and a half billion. So that makes me two and a half billion."

Erdos write many mathematical papers, and co-authors many more. Hoffman explains how leads to the famous Erdos number:

"With 485 co-authors, Erdos collaborated with more people than any other mathematician in history. Those lucky 485 are said to have an Erdos number of 1, a coveted code phrase in the mathematics world for having written a paper with the master himself. If your Erdos number is 2, it means you have published with someone who has published with Erdos. If your Erdos number is 3, you have published with someone who has published with someone who has published with Erdos. Einstein has an Erdos number of 2, and the highest known Erdos number of a working mathematician is 7. The great unwashed who have never written a mathematical paper have an Erdos number of infinity."

Hoffman tells us that there is an Erdos number website:

According to this website, the highest known Erdos number (other than infinity) is 13, but still, most Erdos numbers aren't eight or above.

Hoffman tells us that one of the mathematician's specialties was graph theory. Technically speaking, Lesson 1-4 of the U of Chicago text, when it describes nodes and vertices, is on graph theory. We can think of all mathematicians connected to Erdos as nodes on a large graph, and the Erdos number is the length of the shortest path between any mathematician and Erdos.

The author proceeds to tell us about Ronald Graham -- the mathematician's closest companion. In fact, Graham is a great mathematician in his own right. Graham's number is named after him. In some ways, Graham's number is the opposite of Erdos numbers. While we seek out Erdos numbers that are as small as possible, Graham's number is unimaginably large:

"[Graham] was cited in The Guinness Book of World Records for having come up with the largest number ever used in a proof. The number is incomprehensibly large. Mathematicians often try to suggest the magnitude of a large number by likening it to the number of atoms in the universe or the number of grains of sand in the Sahara. Graham's number has no such physical analogue. It can't even be expressed in familiar mathematical notation, as, say, the number 1 followed by a zillion zeroes. To cite it, a special notation had to be invoked in which exponents are heaped on exponents to form a staggering leaning tower of digits."

Hoffman wraps us the chapter by telling us how uninterested Erdos is in anything not math. He describes a visit to an art museum:

"'We showed him Matisse,' said [mathematician Melvyn] Nathanson, 'but he would have nothing to do with it. After a few minutes we ended up sitting in the sculpture garden doing mathematics.'"

Today is PSAT day across America. In the district whose calendar we're following, there is only the test and breaks today, with no periods or classes at all. Therefore there is no worksheet to post today.

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