problem
Favourite
Picturing Triangular Numbers
Age
11 to 14
Challenge level
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
problem
Favourite
Charlie's delightful machine
Age
11 to 16
Challenge level
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
list
Patterns and Sequences - Short Problems
A collection of short problems on patterns and sequences.
problem
Knockdown
Age
11 to 14
Challenge level
Weekly Problem 51 - 2016
Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?
Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?
problem
Suit Sequence
Age
11 to 14
Challenge level
A pattern repeats every six symbols. What are the 100th and 101st symbols?
problem
Favourite
Picturing Square Numbers
Age
11 to 14
Challenge level
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
problem
Fibonacci Deduction
Age
11 to 14
Challenge level
Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?
problem
Doubly Consecutive Sums
Age
11 to 14
Challenge level
How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?
problem
Favourite
Seven Squares - Group-worthy Task
Age
11 to 14
Challenge level
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
problem
Fruit Line-Up
Age
11 to 14
Challenge level
This grocer wants to arrange his fruit in a particular order, can you help him?
problem
Square grid
Age
11 to 14
Challenge level
Can you work out what fraction of this grid is shaded?
problem
Favourite
Coordinate Patterns
Age
11 to 14
Challenge level
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
problem
Printing Error
Age
11 to 14
Challenge level
Every third page number in this book has been omitted. Can you work out what number will be on the last page?
problem
Tiled Floor
Age
11 to 14
Challenge level
This tiled floor has 109 purple tiles. How many tiles are there altogether?
problem
Favourite
Squares in rectangles
Age
11 to 14
Challenge level
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?
problem
Many Matildas
Age
11 to 14
Challenge level
MatildaMatildaMatil... What is the 1000th letter?
problem
Street Lamps
Age
11 to 14
Challenge level
Walking up a steep hill, I pass 10 equally spaced street lamps. How long do I take to walk from the first lamp to the last?
problem
Favourite
What numbers can we make?
Age
11 to 14
Challenge level
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
problem
Favourite
Triangle Numbers
Age
11 to 14
Challenge level
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
pattern continue?
problem
Favourite
Odds, Evens and More Evens
Age
11 to 14
Challenge level
Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...
article
spaces for exploration
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
problem
Triangular Clock
Age
11 to 14
Challenge level
Trinni rearanges numbers on a clock face so each adjacent pair add up to a triangle number... What number did she put where 6 would usually be?
problem
Favourite
Seven Squares
Age
11 to 14
Challenge level
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
article
Train Spotters' Paradise
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
problem
Night Watchmen
Age
11 to 14
Challenge level
Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?
problem
Favourite
What numbers can we make now?
Age
11 to 14
Challenge level
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
list
Teaching Patterns and Sequences
These articles make the case for teaching in a way that emphasises understanding of underlying structures rather than simply noticing patterns in sequences.
problem
Favourite
Elevenses
Age
11 to 14
Challenge level
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
problem
What a coincidence!
Age
11 to 14
Challenge level
Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?
problem
Favourite
Days and Dates
Age
11 to 14
Challenge level
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
problem
Favourite
Tower of Hanoi
Age
11 to 14
Challenge level
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
problem
Favourite
Beach Huts
Age
11 to 14
Challenge level
Can you figure out how sequences of beach huts are generated?
problem
Favourite
Summing Consecutive Numbers
Age
11 to 14
Challenge level
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?
problem
Favourite
Shifting Times Tables
Age
11 to 14
Challenge level
Can you find a way to identify times tables after they have been shifted up or down?
problem
Favourite
Growing Surprises
Age
11 to 14
Challenge level
Can you find the connections between linear and quadratic patterns?
problem
Favourite
1 Step 2 Step
Age
11 to 14
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?
problem
Pattern Snake
Age
11 to 14
Challenge level
This pattern repeats every 12 dots. Can you work out what a later piece will be?
problem
Hexagon Line
Age
11 to 14
Challenge level
How many hexagons are required for the perimeter of the whole shape to have length 1002cm?
problem
Favourite
Frogs
Age
11 to 14
Challenge level
How many moves does it take to swap over some red and blue frogs? Do you have a method?
problem
Knights and Knaves
Age
11 to 14
Challenge level
Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?
problem
Sliding robot
Age
11 to 14
Challenge level
A robot moves along the number line. Where will it be after 2011 slides?
problem
Even Up
Age
11 to 14
Challenge level
Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?
problem
Expanding Pattern
Age
11 to 14
Challenge level
How many squares are needed to continue this pattern?
problem
How many Rectangles?
Age
11 to 14
Challenge level
By drawing 5 horizontal and four vertical lines, one can form 12 rectangles. What is the greatest number of rectangles that can be formed by drawing 15 lines?
problem
Favourite
Picture Story
Age
14 to 16
Challenge level
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
problem
Series Sums
Age
14 to 16
Challenge level
Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.
problem
Favourite
Double Trouble
Age
14 to 16
Challenge level
Simple additions can lead to intriguing results...
problem
Favourite
Attractive Tablecloths
Age
14 to 16
Challenge level
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
problem
Collatz 13
Age
14 to 16
Challenge level
If a number is even, halve it; if odd, treble it and add 1. If a sequence starts at 13, what will be the value of the 2008th term?
problem
Ordered Sums
Age
14 to 16
Challenge level
Let a(n) be the number of ways of expressing the integer n as an
ordered sum of 1's and 2's. Let b(n) be the number of ways of
expressing n as an ordered sum of integers greater than 1. (i)
Calculate a(n) and b(n) for n<8. What do you notice about these
sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove
your conjectures.
problem
Collatz-ish
Age
14 to 16
Challenge level
A sequence is generated using these rules. For which values of n is the nth term equal to n?
problem
Favourite
Painted Cube
Age
14 to 16
Challenge level
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
problem
Trolley Park
Age
14 to 16
Challenge level
In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?
problem
12345
Age
14 to 16
Challenge level
Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?
problem
Diagonals
Age
14 to 16
Challenge level
How many diagonals does a regular icosagon (20 sides) have?
problem
Below 400
Age
14 to 16
Challenge level
Can you work out which number will appear directly below 400 in this pattern?
problem
Alternating Sum
Age
14 to 16
Challenge level
Given that the number 2008 is the correct answer to a sum, can you find n?
problem
Favourite
Mystic Rose
Age
14 to 16
Challenge level
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
problem
Favourite
Partly Painted Cube
Age
14 to 16
Challenge level
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
problem
Favourite
A little light thinking
Age
14 to 16
Challenge level
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
problem
Newspaper Sheets
Age
14 to 16
Challenge level
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?
problem
Favourite
Steel Cables
Age
14 to 16
Challenge level
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?