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Proximity

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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Kite in a Square

Can you make sense of the three methods to work out the area of the kite in the square?

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Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Take some triangular numbers. Multiply each one by eight and add one.There is an interesting pattern - can you spot it? Test your conjecture with some other triangular numbers.

Can you prove that the pattern continues for any triangular number?

If you are finding it tricky to construct a proof, an activity is available here which allows you to put a proof we have written in the correct order. Alternatively, look at the hint for a visual way of thinking about this problem.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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NRICH is part of the family of activities in the Millennium Mathematics Project.