There are **106** NRICH Mathematical resources connected to **Trial and improvement**, you may find related items under Using, Applying and Reasoning about Mathematics.

A cinema has 100 seats. Is it possible to fill every seat and take exactly £100?

Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

Can you use the information to find out which cards I have used?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?

You have two sets of the digits 0 – 9. Can you arrange these in the five boxes to make four-digit numbers as close to the target numbers as possible?

Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Use the information about Sally and her brother to find out how many children there are in the Brown family.

There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. Can you use the information given to find out how many eggs are in each basket?

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.

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.

Amy's mum had given her £2.50 to spend. She bought four times as many pens as pencils and was given 40p change. How many of each did she buy?

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

There are lots of different methods to find out what the shapes are worth - how many can you find?

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

Find at least one way to put in some operation signs (+ - x ÷) to make these digits come to 100.

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

What do you notice about these squares of numbers? What is the same? What is different?

Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Make one big triangle so the numbers that touch on the small triangles add to 10.

Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.

Use these four dominoes to make a square that has the same number of dots on each side.

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?