# Test your brain power

## Hard

Milly Miffen made a muffin more than Molly’s mother made, and Milly Miffen’s mother made a muffin more than Molly made; and Milly, Molly’s mother, Molly, and Milly’s mother made 50 muffins, but Milly and Molly’s mother made four muffins more than Molly and Milly’s mother made.

So murmur now how many muffins Milly made.

JAH Hunter in Fun with Figures, submitted by Robb Gosine, postgraduate student

## Very hard

Is it possible to find five points in an equilateral triangle of side 10cm so that each pair of points is more than 5cm apart?

Dr Lynda White, Department of Mathematics

## See the answer

### See the answer

Answer: No. Divide the triangle into four smaller equilateral triangles each of side 5cm. One of these four triangles must contain at least two of the five points and two such points are at most 5cm apart.

## Fiendish

My maths teacher drew three arbitrary circles on the whiteboard. She then drew the common external tangents to a pair of the circles, which, of course, intersected. She challenged me to demonstrate that the intersection points of the three tangent pairs were collinear.

Professor Myron L. Good, submitted by David Lloyd Owen (BSc Physics 1972)

## See the answer

### See the answer

Answer: Generalise the problem to three dimensions: circles become spheres; tangent pairs become cones.

Now, supposing the spheres are centred in a horizontal plane, bring in two planes – one from above, one from below – until they touch the three spheres. Both planes include the vertices of the cones (the intersection points of the tangents), and, necessarily, the planes intersect along a straight line.

The intersection points must, therefore, be along this line.

## How to enter:

Senders of correct solutions for two or more of the puzzles will be entered into a prize draw to win a copy of Superhuman Innovation: Transforming Business with Artificial Intelligence by Chris Duffey.

Winners' names will be in Imperial 50 in May 2021, and solutions published on this page.

To enter, please email imperialmagazine@imperial.ac.uk.

Entries are now closed.

Congratulations to the first ten respondents who contacted us with two or more correct solutions from the issue:

• Alexander Dean (MSci Physics with Theoretical Physics 2018)
• Jennie Yang (Biochemistry 2006)
• Alan Cleugh (Chemical Engineering 1960)
• Kathryn Sayer (MEng Mechanical Engineering 2016)
• Xin Yi (Mathematics 2012)
• Mike Grannell (Mathematics 1967, Mathermatics PhD 1967-70)
• AJ Denham (MEng Mining Engineering 1987)
• Chris Johnson (Physics 1973)
• Jason Klimach (Mathematics 2008)
• Nick Grogan (MSc Mathematics 1978)