Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and. . . .
In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
Make some intricate patterns in LOGO
Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?
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?
I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?
Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.
Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...
In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Have a go at this 3D extension to the Pebbles problem.
Can you find a way to identify times tables after they have been shifted up?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
A introduction to how patterns can be deceiving, and what is and is not a proof.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?
A story for students about adding powers of integers - with a festive twist.
If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?
This article for teachers describes the exchanges on an email talk list about ideas for an investigation which has the sum of the squares as its solution.
Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
Investigate what happens when you add house numbers along a street in different ways.
There are lots of ideas to explore in these sequences of ordered fractions.
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
How many different sets of numbers with at least four members can you find in the numbers in this box?
According to an old Indian myth, Sissa ben Dahir was a courtier for a king. The king decided to reward Sissa for his dedication and Sissa asked for one grain of rice to be put on the first square. . . .
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?
July 1st 2001 was on a Sunday. July 1st 2002 was on a Monday. When did July 1st fall on a Monday again?
Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
Find the next number in this pattern: 3, 7, 19, 55 ...
Investigate these hexagons drawn from different sized equilateral triangles.
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.
In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?
An environment which simulates working with Cuisenaire rods.
There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.
Liitle Millennium Man was born on Saturday 1st January 2000 and he will retire on the first Saturday 1st January that occurs after his 60th birthday. How old will he be when he retires?
These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?