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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.)
Stage: 5 Short Challenge Level: 
Here are 16 propositions involving a real number $x$:
| $x\int^x_0 ydy < 0$ | $x> 1$ | $0< x< 1 $ | $x^2+4x+4 =0$ |
| $x=0 $ | $\cos(x/2)> \sin(x/2)$ | $x> 2$ | $x=1$ |
| $2\int^{x^2}_0ydy> x^2 $ | $x< 0 $ | $x^2+x-2=0$ | $x=-2 $ |
| $x^3> 1$ | $|x|> 1$ | $x> 4$ | $\int^x_0 \cos y dy =0$ |
[Note: the trig functions are measured in radians]
By choosing $p$ and $q$ from this list, how many correct mathematical statements of the form $p\Rightarrow q$ or $p\Leftrightarrow q$ can you make?
It is possible to rearrange the statements into four statements $p\Rightarrow q$ and four statements $p\Leftrightarrow q$. Can you work out how to do this?
NOTES AND BACKGROUND
Logical thinking is at the heart of higher mathematics: In order to construct clear, correct arguments in ever more complicated situations mathematicians rely on clarity of language and logic. Logic is also at the heart of computer programming and circuitry. To find out more, look at the ideas surrounding the Adding Machine problem and related set of activities.
NRICH is part of the family of activities in the Millennium Mathematics Project.