## Robot maze puzzle

The puzzle above has a robot (marked with an arrow pointing up, or “north”) that you can control with a set of command cards that either move the robot forwards a set number of spaces (the + numbers) or backwards (the – numbers). After using a command card, the card is used up and can’t be used again.

If the robot hits either the border of the grid or one of the black spaces, the robot stops moving and any remaining steps on the command card being used are ignored.

Landing on one of the spaces marked with circles causes your robot to turn 90 degrees. (That is, if the robot faces north it turns east, if east it turns south, if south it turns west, and if west it turns north.) The robot starts facing north. Can you get the robot to the star?

## Counting puzzle from the first US puzzle championship

My students had fun with this one today. Part d is what showed up in the actual championship.

One night, in an attempt to combat insomnia, you begin counting (1, 2, 3, …), but you decide to do it digit by digit. As you go along, for example, the 15th digit you count is the 2 of the number 12.
1 2 3 4 5 6 7 8 9 10 11 12

a.) What’s the 50th digit you count?
b.) What’s the 100th digit you count?
c.) What’s the 1000th digit you count?
d.) What’s the 1,000,000th digit you count?

## 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: