### 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.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Largest Product

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

Various sets of numbers add together to give a sum of 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 greatest product that can be made from whole numbers that add up to 100?

Try using different starting numbers.

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