Creating and manipulating expressions and formulae

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

Can you verify and generalise these equations?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain why this works.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

Find the five distinct digits N, R, I, C and H in the following nomogram

Can you find a rule which connects consecutive triangular numbers?