# Working Systematically - Lower Secondary

Working Systematically is part of our Thinking Mathematically collection.
Scroll down to see our complete collection of problems that require you to work systematically, or explore the two sub-collections focussing on important aspects of systematic working.

### Noticing Patterns

The key to solving these problems is to notice patterns or properties. Organising your work systematically allows you to notice what might not otherwise be obvious.

### Finding All Solutions

These problems challenge you to find all possible solutions. One of the best answers to "How do you know you have found them all" is to be able to say "I worked systematically!"

### Two and Two

##### KS 3 Challenge Level:

How many solutions can you find to this sum? Each of the different letters stands for a different number.

### Summing Consecutive Numbers

##### KS 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Isosceles Triangles

##### KS 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### ACE, TWO, THREE...

##### KS 3 Challenge Level:

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

### American Billions

##### KS 3 Challenge Level:

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

### M, M and M

##### KS 3 Challenge Level:

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

### 1 Step 2 Step

##### KS 3 Challenge Level:

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

### Can They Be Equal?

##### KS 3 Challenge Level:

Can you find rectangles where the value of the area is the same as the value of the perimeter?

### Pick's Theorem

##### KS 3 Challenge Level:

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

### Sticky Numbers

##### KS 3 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

##### KS 3 Challenge Level:

How many different symmetrical shapes can you make by shading triangles or squares?

### Shifting Times Tables

##### KS 3 Challenge Level:

Can you find a way to identify times tables after they have been shifted up?

### Peaches Today, Peaches Tomorrow....

##### KS 3 Challenge Level:

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

### Charlie's Delightful Machine

##### KS 3 & 4 Challenge Level:

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

### Nine Colours

##### KS 3 Challenge Level:

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

### What's Possible?

##### KS 4 Challenge Level:

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

### A Long Time at the Till

##### KS 4 & 5 Challenge Level:

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

### Product Sudoku

##### KS 3 Challenge Level:

The clues for this Sudoku are the product of the numbers in adjacent squares.

### Add to 200

##### KS 3 Challenge Level:

By selecting digits for an addition grid, what targets can you make?

### Gabriel's Problem

##### KS 3 Challenge Level:

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?