# Tag Archives: ramsey theory

## 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.

Filed under Accessible, Puzzle

## Tie Fighters

Noughts and crosses (tic-tac-toe) is quite a boring game. The two main reasons for this are:

1. Most games end in a draw or tie.
2. The optimal strategy is too obvious (the first player wants to start in the middle).
With an ulterior mathematical motive in mind, I’d like to introduce you to two games that avoid the first complaint. Whether they address the second is up to you: I’d argue that only Game B, called Sim, does.
Game A
Let’s call the game triangle-tac-toe. Draw a grid in the shape of a right-angled isosceles triangle whose sides are five squares across (giving 15 squares in total). A board-drawing hint: draw five rectangles.
Starting grid
Take turns to add noughts or crosses as in traditional tic-tac-toe. A player wins when three of their marks form the corners of a right-angled isosceles triangle of any size in the same orientation as the board (the corners may be touching or spread out).
Some of the possible ways for Crosses to win.
Game B (Sim)
Draw six points in a hexagonal arrangement. Optionally, lightly join each possible pair of points (a total of 15 lines) with dotted lines—don’t worry whether three cross lines in the middle or not, but double check that each point has five lines coming out of it.
Sim starting layout
Players pick their favourite colours (we’ll use the traditional Red and Blue), and take turns drawing a straight line in their chosen hue between two points (that haven’t already been connected). A player loses if they form a triangle of their colour between any three of the hexagon’s vertices.
End of a sample game—Red loses due to the highlighted triangle
Animation of the sample game above (click if not playing)