Monthly Archives: November 2011

Getting into norms

This blog is named “Out of the Norm”, and the logo is the list of axioms that a norm satisfies (with some extra graffiti), but there’s a fair chance that it’s just mathematical gobbledegook to you. So what is a norm? At its heart, a norm is essentially a measure of distance, though not necessarily a spatial distance. Here are a couple of examples to give you some informal sense of what a norm is, and how they might be used.

Distance in space

If I asked you to go find some buried treasure 3 units of distance East, and 4 units North, how far would you have to travel? That is, from the origin (0, 0) to the point (3, 4) on the graph or ‘map’ below?

It’s not a trick question: the distance as the crow flies, by Pythagoras’ theorem, is simply \sqrt{3^2 + 4^2}=5.

But what if it were a trick question? Let’s say you were actually a taxi driver in some Manhattan-like city with square blocks (each a single unit in size), and were taking a passenger three blocks East and four blocks North (perhaps to the buried treasure, if you insist). Then the shortest route would be seven units long (7=3+4; there are many different shortest routes).

Two of the shortest routes

So now you’ve seen two different notions of distance in 2-dimensional space: the first very familiar; the second probably less so. Just to reinforce the point, you can do the same thing in 3-dimensional space. The distance as the crow flies directly to a point 5 units above the unnecessary treasure, that is, the point (3, 4, 5), is approximately seven units: \sqrt{3^2+4^2+5^2}=\sqrt{50}. However, our taxi cab driver has to drive around the buildings and then take some sort of car-lift, and travels 3+4+5=12 units.

Diagram not at all to scale.

You can extend this to as many spatial dimensions as you want, though as far as I know it’s impossible for humans to imagine a diagram with the point (3, 4, 5, 2) hovering somewhere. But that doesn’t mean the notion is useless.

Guess the age of the lecturers

The next example has been used by one of my former colleagues to introduce norms to his students. Firstly, he gets the students to each write down a guess of the ages of all the lecturers in the maths department, and writes the guesses up on the board. Then he reveals the true ages of the lecturers, and asks the students how they should decide who has won. Here’s some completely made-up numbers:

Let’s say there are four lecturers: a PhD student, a lecturer, a senior lecturer, and a professor, who are 25, 35, 45 and 60 years old respectively. We’ll write this as the vector (25, 35, 45, 60). Now the three students make their guesses: Adam guesses in order 27, 36, 55 and 66: again, write this as a vector (27, 36, 55, 66). Beth guesses (26, 35, 53, 51), completely misjudging the age order of the older pair. And, because there’s always one in every class, Charlie guesses (27.2, 34, 56.4, 62.4), which reminds us that we needn’t work only with whole numbers.


The students decide to judge the results Continue reading

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Algebraic Linesweeper

[If you want to dive straight into Algebraic Linesweeper, here’s a printable PDF with rules.]

A friend of mine, Jak Marshall, created a pen-and-paper puzzle game called Linesweeper. Here are the rules:

  • Draw a single, closed, continuous loop in the empty cells of the grid which never crosses itself or branches.
  • The number clue in a cell indicates how many of the 8 adjacent cells are part of the loop (as in the classic computer game Minesweeper); for instance, a ‘0’ means that none of the adjacent cells are part of the loop.
  • The loop may not enter a cell with a number.
  • The loop may run horizontally or vertically (not diagonally) between centres of adjacent cells (that is, parallel to the grid lines).
  • The loop does not need to pass through all the unoccupied cells.
  • Each puzzle should have a unique solution.
Here’s an example, with its solution immediately below:
Linesweeper example
Linesweeper solved example
You can find this example and at least 20 more online Java puzzles at the German puzzle site janko.at. [Edit: Andrea Sabbatini has included Linesweeper (“the looping minesweeper”) into her 56 Logic Game Time Killers puzzle pack free for the iPhone and iPad. Linewsweeper works well on a touch screen.] Cross+A has even included Linesweeper in their commercial puzzle solver.

I think it’s fun and relaxing, and more satisfying to progress and finally connect the loop than fill in the final cells of a Sudoko puzzle. You can also design puzzles that use a bit of simple graph theory: if there is an area that your line visits with only three possible routes in and out, it can only use an even number (two) of these (the loop is in some sense a Eulerian circuit).

Algebraic Linesweeper

I’d like to introduce a variant of Linesweeper that I created. Instead of giving the number clues for a puzzle in a straightforward manner, I only give some letters that stand in for them. To be more specific:
  • Each letter corresponds to a unique number between 0 and 8.
  • Within a puzzle, different letters must correspond to different numbers.
  • Each puzzle should still have a unique solution!
I’ve made a handy one-page printable pdf file of six Algebraic Linesweeper puzzles, with the rules. The result, at first glance seems impossible. To convince you it is, and check that you’ve understood, here’re two tiny practice puzzles:
The “Elementary” puzzle illustrates the point I was making about Eulerian circuits above: you can view the puzzle as a more discrete version of the following diagram inspired by the Bridges of Königsberg.
Here’s the six actual puzzles from the above pdf in an approximate order of difficulty. Obviously, the choice of letters is slightly arbitrary, so I thought I would choose them based on some theme. If I can spell out words, like below, then it gives a convenient title to identify each puzzle:

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Blitz Krieger

I thought the following story would make a good question to add flavour to some first-year undergraduate Number Theory courses. It’s relevant, interesting, well-documented, and shows how a certain amount of scientific literacy can be extremely useful when evaluating bogus claims. Here’s an extract from a 1938 TIME Magazine article (££)

In 1937 A.D., a German-Jewish mathematician named Samuel Isaac Krieger, who was taking a mineral bath near Buffalo, N. Y., suddenly leaped out, rushed naked into the adjoining room, began to scribble figures. He thought he had discovered something too: a solution to the equation given in Fermat’s last theorem.

Krieger claimed to have found some integer n > 2 such that the equation 1324^n + 731^n =1961^n held, but refused to reveal the value of n.  A journalist quickly proved this to be false. Can you see how?

Obviously, Andrew Wiles only proved Fermat’s Last Theorem (that there are no integers a,b,c > 0 such that a^n + b^n = c^n for n > 2) to be true in full generality after 1994, so he couldn’t use that. While it was known at the time that the theorem was true for exponents including n=3,4,5,7, the journalist only used elementary mathematics and did not need to use any specific knowledge about the theorem.

Solution after the break. Also, how not to solve FLT, and more on Krieger.

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