### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Calendar Capers

Choose any three by three square of dates on a calendar page...

### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

# Largest Product

##### Stage: 3 and 4 Challenge Level:

Here are some different ways in which we can split 100:

• $30 + 70 = 100$
• $20 + 80 = 100$
• $21 + 56 + 23 = 100$
• $10 + 10 + 10 + 10 + 10 + 10 + 20 + 20 = 100$

The products of these sets are all different:

• $30 \times70 = 2100$
• $20 \times 80 = 1600$
• $21 \times 56 \times23 = 27048$
• $10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 20 \times 20 = 400000000$

What is the largest product that can be made from whole numbers that add up to 100?

Choose another starting number and split it in a variety of ways.
What is the largest product this time?

Can you find a strategy for splitting any number so that you always get the largest product?