Partitions

Introduction

Write 10 as the sum of some (it can be more than two) positive integers, e.g. 7 + 3 = 10.

Find the product of the integers that sum to 10, e.g. 7 × 3 = 21.

How many different ways can we add positive whole numbers to make 10?

What are the products of these sets?

What is the highest product that you can make?

Further Questions and Challenges

Explore for other starting numbers, try to develop some conjectures e.g. look at how each number is broken down: 4 can be broken into 2 x 2 which doesn’t improve the total, whereas 5 can be 2 x 3 (= 6) which does.

What happens to multiples of 3, one more than a multiple of 3, one less than a multiple of 3; assume that the numbers are sufficiently large?