Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Impossible Triangles?

Stage: 5 Challenge Level:

You are given a limitless supply of triangular jigsaw pieces of type T1, T2, T3, T4 and T5

Using these pieces, you need to try to make larger triangular shapes without any overlap.

Three triangle shapes are shown in the picture below. Is it possible to create two-coloured triangles of these shapes without moving the pieces already placed?

Suppose next that you can move the pieces and choose two triangle types to work with. Which pairs of shapes can be used to make larger equilateral triangles? Which pairs of shapes can be used to make larger 30-60-90 triangles? See the video clip for a discussion on this part of the problem

If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.

NOTES AND BACKGROUND

This problem is all about either finding solutions or proving that there is no solution for any size of triangle shape.

For small triangle shapes, it is easy to check all possible configurations of pieces to check whether a solution exists. For larger triangle shapes the number of combinations of pieces gets larger extremely rapidly, and quickly reaches the point at which a check of all of the combinations is impossible, even on a supercomputer. Even if we have checked a large number of jigsaw sizes and found no solution, this does not necessarily mean that we cannot find a solution for a larger triangle shape. To find a solution, you often need to mine the depths of your cunning and ingenuity. To show that a solution does not exists you often have to use the concept of proof by contradiction .

Proving that a solution does not exist is often much easier than proving that a solution does exist. Interestingly, if a solution can be found, then it is usually very quick to check that the solution is correct, although finding the solution in the first place might be exceptionally difficult.This behaviour underlies the notion of the mysteriously titled 'P vs NP' problem, the solution of which will earn the solver \$1,000,000

Ideas concerning proof are discussed in the fascinating article Proof: A Brief Historical Survey