Working Systematically
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
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Summing Consecutive Numbers
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Isosceles Triangles
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?
American Billions
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
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
1 Step 2 Step
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Pick's Theorem
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
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Shady Symmetry
How many different symmetrical shapes can you make by shading triangles or squares?
Shifting Times Tables
Can you find a way to identify times tables after they have been shifted up or down?
Peaches Today, Peaches Tomorrow...
A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?
Charlie's Delightful Machine
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Nine Colours
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Consecutive Negative Numbers
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Cuboids
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Weights
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Sociable Cards
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
Triangles to Tetrahedra
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Fence It
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Where Can We Visit?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Cinema Problem
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
Add to 200
By selecting digits for an addition grid, what targets can you make?
Squares in Rectangles
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
Gabriel's Problem
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Multiples Sudoku
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Number Daisy
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
Special Numbers
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Ben's Game
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
Shopping Basket
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
Product Sudoku
The clues for this Sudoku are the product of the numbers in adjacent squares.
Difference Sudoku
Use the differences to find the solution to this Sudoku.
LCM Sudoku
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Warmsnug Double Glazing
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?