### Iff

Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

### Take Ten Sticks

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

### Reasonable Algebra

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

# Mystic Rose

##### Stage: 4 Challenge Level:

A Mystic Rose is a beautiful image created by joining together points that are equally spaced around a circle.

Move the sliders below to see how a Mystic Rose can be constructed. You can change the number of points around the circle.

Can you describe how to construct a Mystic Rose?

Alison and Charlie have been working out how many lines are needed to draw a 10 pointed Mystic Rose.

Alison worked out $9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45$.

Charlie worked out $\frac{10 \times 9}{2} = 45$

Can you explain how each method relates to the construction of a 10 pointed Mystic Rose?

How would Alison work out the number of lines needed for other Mystic Roses?
How would Charlie work them out?

Whose method do you prefer?

How many lines are needed for a 100 pointed Mystic Rose?

Could there be a Mystic Rose with exactly 4851 lines?
Or 6214 lines?
Or 3655 lines?
Or 7626 lines?
Or 8656 lines?

How did you decide?

Final Challenge

In a chess tournament every contestant is supposed to play exactly one game against every other contestant.
However, contestant A withdrew from the tournament after playing only ten games, and contestant B withdrew after just one game.
A total of 55 games were played.

Did A and B play each other?

You may wish to try the problems Picturing Triangle Numbers and Handshakes. Can you see why we chose to publish these three problems together?

You may also be interested in reading the article Clever Carl, the story of a young mathematician who came up with an efficient method for adding lots of consecutive numbers.