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Can you make a tetrahedron whose faces all have the same perimeter? A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground? Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Rational Integer

##### Age 14 to 16 ShortChallenge Level

Answer: six values of $n$ ($-3, -1, 0,2,3,5$)

Using factors
$n-1$ is a factor of $n+3$ means $n$ is quite small because otherwise $n-1$ is too close to $n+3$

 $n$ $n-1$ $n+3$ fit? 2 1 5 yes 3 2 6 yes 4 3 7 no 5 4 8 yes 6 5 9 no and it won't work for larger numbers because 5 is more than half of 9

Or if $n$ can be negative:

 $n$ $n-1$ $n+3$ fit? 0 $-$1 3 yes $-$1 $-$2 2 yes $-$2 $-$3 1 no $-$3 $-$4 0 yes $-$4 $-$5 $-$1 no from now on, the 'size' of $n-1$ will be greater than the 'size' of $n+3$ so we won't get any more fits

Using algebra
$\frac{n+3}{n-1} = \frac{n-1}{n-1} + \frac{4}{n-1} = 1 + \frac{4}{n-1}$. Thus $\frac{n+3}{n-1}$ is an integer if and only if $n-1$ divides exactly into $4$. The values of $n$ for which this is true are $-3, -1, 0,2,3,5$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.