## The origami proof that the square root of 2 is irrational

Fold from upper right to lower left (colors added to both sides of the paper for clarity):

Follow up with one more fold:

Voila, a proof that the square root of 2 is irrational.

(Mind you, there is some reasoning involved, but what’s the fun in giving that away? Start with “if the square root of 2 is rational, then there is some isosceles right triangle where the sides are the smallest possible integers.”)

This proof first appears in slightly different form in 1892. The paper folding version is from Conway & Guy in 1996.

## Guess the purpose of this patent

What’s this for? Circa 1878:

## Visual Algebra?

In recent discussion about Bret Victor’s Kill Math project Ben Blum-Smith brought up the books Visual Complex Analysis and Visual Group Theory which as he puts it “all arguments are geometric and illustrated by diagrams”. (I’m not familiar with the latter, but Visual Complex Analysis is fantastic and I highly recommend it.)

I feel like these sorts of books will eventually create a revolution in upper-level mathematics — I’m eagerly awaiting someone to write Visual Linear Algebra — but could we re-conceive lower level mathematics in the same way?

By Visual Algebra I’m not meaning graphs, I’m meaning the more mundane symbolic “solve for x” manipulation.

Solve for x: 2x + 3 = 5.

In the same vein as my puzzle equivalent to solving a quadratic, solid lines mean multiply, dotted lines mean add.

Solving for the highlighted circle is equivalent to solving for x in 2x + 3 = 5.

(I swear I have seen something closely resembling this elsewhere for equations, and I think it even has a buzzword attached — anyone know?)

I originally thought of these sorts of puzzles as a gentle introduction to the topic, but would it be possible to integrate this kind of visual-symbolic thinking in every part of an algebra course?

ADD: Here’s an image where the puzzle is closer in look to the equation:

This sort of thing is risky because rather than applying inverses and so forth students may make it a general method to draw circles and arrows everywhere.

## Induction puzzles from the 2003 World Puzzle Championship

The vast majority of the problems in the World Puzzle Championship are of the same ilk as from the most recent championship in Turkey. Every once in a while an induction puzzle sneaks on, the kind where one looks for a pattern and guesses the rule or fills in the missing pieces.

I’ve had these puzzles lurking about since the 2003 championship in the Netherlands occurred, and I’ve never been able to solve four of them. I figure at seven years old it is high time to get them off my queue.

First off, a set called Common Touch based off of Bongard problems. I know the answers to only the first two.

In each of the three puzzles, 4 puzzlers in the YES group all share an unusual property, which none of the names in the NO group have. For each puzzle, pick one of the 8 names from the answer list that shares the property in the YES group. Note that the answers have nothing to do with the people themselves, only the names.

Puzzle 1:

YES: LASLO MERO, CLAUDE DESSET, NECMIYE OZAY, ANDREAS BOLOTA

NO: KAROLY KRESZ, METIN BALCI, PAVEL KALHOUS, ROGER BARKAN

Puzzle 2:

YES: CLAUDE DESSET, NIELS ROEST, LASZLO OSVALT, ANNICK WEYZIG

NO: DAVID SAMUEL, DELIA KEETMAN, TETSUYA NISHIO, RON OSHER

Puzzle 3:

YES: PETR NEPOVIM, BIRGIT ROSENTHAL, PAVEL KALHOUS, DARIUSZ GRABOWSKI

NO: TIM PEETERS, HUSNU SINCAR, ALEXANDRU SZOKE, EMERIC LORINCZ

Answer list: JOHN WETMILLER, JAN LAM, HANS EENDEBAK, ULRICH VOIGT, ZACK BUTLER, JAN FARKAS, ROBERT BABILON, SILKE RITTER

Out of these “fill in the question mark” problems, I have solved exactly zero:

## Blogroll Shuffle II, personal news

Some blogs have a “kitchen sink” approach to collecting blogroll links, and I often come to them when I’m looking for something new (JD2718′s is good) but I approach my blogroll with a “museum curator” standpoint, which means sometimes I have to rotate the exhibits. I have put in a fresh batch of 7 and added the old links to the annotated blogroll. (If yours has been left out, please don’t interpret it badly. This is just a selected assortment out of many very good blogs.)

I also have added a “puzzles” section, which is mostly filled with Nikoli-style puzzles. If you don’t know what Nikoli-style means, please try the links: you are missing an entire world! I would recommend starting with Grant Fikes. One the blogs (Mokauni’s) is in Japanese; save it until you are fully comfortable with the standard puzzle types.

I would also like to take the opportunity to mention I am taking a leave of absence next year to join the faculty of the University of Arizona. I will be working with elementary and middle school teachers teaching them how to teach mathematics, and developing an online curriculum to do the same.

## Josh Giesbrecht takes on factoring puzzles

Last year I posted about a puzzle equivalent to factoring a quadratic equation:

(Solid lines mean multiply, dotted lines mean add.)

Josh Giesbrecht has produced some lovely full sheets worth of these puzzles.

## Philadelphia Sudoku alleged cheater update

The competetor suspected of cheating at the 2009 Philadelphia Inquirer Sudoku National Championship is going to be retested.

Also, the picture has been positively identified by Phil Irwin:

Irwin played Varshavsky in the National Open in June 2006, two weeks before the World Open. Irwin wrote by e-mail that players of their level usually start with simple opening moves and wait for an opponent’s blunder, but “during our game he played some very bizarre opening moves and then eventually quickly finished me off with a very sophisticated combination. He arrived late to the game and kept his neck cocked at the same angle for long periods. Later I wondered if he had a camera in his stocking cap, which he wore along with a heavy sweater in Las Vegas in June.”

(Tip of the hat to Thomas Snyder.)

## Selections from the instructions for the 18th World Puzzle Championship

The 18th World Puzzle Championship starts today in Turkey and runs through this week.

Just to give a taste of how things will go, here are three selections from the full instructions for this year, available from the official website.

Circle Division

Draw given number of straight lines in the circle so that the sum of the numbers in all formed regions are the same. Lines should start and end on the circle perimeter and all formed regions should contain numbers.

Star Battle

Place exactly two stars in each row, each column and each outlined area. Stars have the size of one cell and cannot touch each other, not even diagonally.

Hang Up

Hang a rectangle (possibly a square) from its corner on each nail. All rectangles should have different perimeter lengths (not necessarily different areas) and their center of gravity should always be on the line going down from that nail. Rectangles cannot touch each other, not even diagonally.

As an extra bonus, here are the two Star Battle puzzles from Turkey’s own 2009 Puzzle Championship:

## Cheating at the 2009 Philadelphia Inquirer Sudoku National Championship

This story was originally broken — and I mean broken, not just copied from elsewhere — by Thomas Snyder, 2-time winner of the World Sudoku Championship. The story is ongoing; perhaps you can help catch a cheat?

The 2009 Philadelphia Inquirer Sudoku National Championship occurred last week. It was done in three common rounds (with everyone solving puzzles in the same room), where the top 3 scores went on to a final round (depicted above).

The winners:

1. Tammy McLeod [in center of photo], Los Angeles (\$10,000 and a seat on the U.S. team at the World Sudoku Championship in April in Philadelphia). She came in third last year. She finished the final round in 7 minutes, 41 seconds.

2. Thomas Snyder [in left of photo], Palo Alto, Calif. (\$4,000). He finished in second place last year, and is a two-time world champion.

3. Eugene Varshavsky [in right of photo], Lawrenceville, N.J. (\$3,000). He joined the competition during walk-up registration yesterday morning.

Thomas Snyder finished the final puzzle in 4:14, but made a transposition error allowing Tammy McLeod the win. The Inquirer has many more details.

The person on the right, Eugene Varshavsky, entered the competition in highly irregular circumstances. He skipped the first and second rounds altogether, arriving late. He then proceeded to finish the 3rd round in blazing time, qualifying him for the final. This is his grid at the end of the competition:

As Thomas Snyder writes:

It has 2 observable placements in it, both in row 5, and a suggestion that the 9 in R1C3 may be there too (eta: confirmed from other images now). It is however not the most focused image and does not tell how this grid got to this state, if erasing happened, etc. Still, having this for 8 minutes of work on the puzzle after demolishing 3 hard ones in 12-13 minutes to qualify is simply not possible.

Eugene wore the hood you see in the first picture the entire competition. There is only one picture of him with the hood down:

He gave his residence as Lawrenceville, N.J., but no one by that name is listed in the town, and efforts to discover his actual residence have turned up a trail of dead ends.

A LexisNexis search revealed that a Eugene Varshavsky in 2007 had given his residence as an address in Ewing, not far from Lawrenceville. But that address leads only to HB Machines, where proprietors said they knew of no such person.

This may be the second time this particular cheat may have shown up at a competition. Thomas Snyder again, regarding a “Varshavsky” at the World Open Chess Tournament:

In 2006, for example, a suspected incident of cheating occurred in the World Open Chess Tournament. Against Grandmaster Smirin, a relatively unknown player wearing a hat the whole time performed well beyond expectations and ranking to beat the Grandmaster. After some suspicion was raised, this unknown disappeared to a bathroom where after ten minutes he was searched and nothing was found. Under closer watch, without the possibility of using unallowed assistance, the performance of this player returned to more expected levels and he lost the following matches, coming nowhere near to the mastery he had demonstrated earlier.

Because of the mystery surrounding Varshavsky, this story is still developing. Was anyone who is reading this at the 2006 World Open? Does anyone recognize the man in the picture?