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:

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?