Are these statements always true, sometimes true or never true?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Delight your friends with this cunning trick! Can you explain how it works?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Here are two kinds of spirals for you to explore. What do you notice?

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

This activity involves rounding four-digit numbers to the nearest thousand.

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. . . .

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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 dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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 NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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. . . .

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Can you find all the ways to get 15 at the top of this triangle of numbers?

What happens when you round these three-digit numbers to the nearest 100?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

This task follows on from Build it Up and takes the ideas into three dimensions!