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There are **109** NRICH Mathematical resources connected to **Trial and improvement**, you may find related items under Thinking mathematically.

Problem
Primary curriculum
Secondary curriculum
### Going to the Cinema

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

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Five Steps to 50

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All Change

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fifteen Cards

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Jumping Squares

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.

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factor Track

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Area and Perimeter

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Inky Cube

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?

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Strike it Out

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

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Coded Hundred Square

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two-digit Targets

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Four-digit Targets

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Treasure Hunt

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

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Find the Difference

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### In the Bag

Can you guess the colours of the 10 marbles in the bag? Can you develop an effective strategy for reaching 1000 points in the least number of rounds?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Root Hunter

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factor-multiple Chains

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sums of Pairs

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?”

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magic Potting Sheds

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?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fence It

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sending a Parcel

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Tall Tower

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Many Miles to Go?

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Brown Family

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Eggs in Baskets

There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. How many eggs are in each basket?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cinema Problem

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fitted

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cycling Squares

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One Wasn't Square

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plenty of Pens

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fractions in a Box

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's it Worth?

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

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Twenty Divided Into Six

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Make 100

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Zios and Zepts

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Watch the Clock

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's in a Name?

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Highest and Lowest

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Prison Cells

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Heads and Feet

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One to Eight

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Old Am I?

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Grouping Goodies

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Puzzling Sweet Shop

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?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Chain of Changes

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One Big Triangle

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Round Up

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.

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 4 Dom

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

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Domino Square

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Big Dog, Little Dog

Woof is a big dog. Yap is a little dog. Emma has 16 dog biscuits to give to the two dogs. She gave Woof 4 more biscuits than Yap. How many biscuits did each dog get?

Age 5 to 7

Challenge Level