Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Can you describe this route to infinity? Where will the arrows take you next?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Delight your friends with this cunning trick! Can you explain how it works?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
Can you explain how this card trick works?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
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?
Can you explain the strategy for winning this game with any target?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Can you find sets of sloping lines that enclose a square?
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. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
A game for 2 players
It would be nice to have a strategy for disentangling any tangled ropes...
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
Can you tangle yourself up and reach any fraction?
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
It starts quite simple but great opportunities for number discoveries and patterns!
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Charlie has moved between countries and the average income of both has increased. How can this be so?
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?