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There are **445** NRICH Mathematical resources connected to **Working systematically**, you may find related items under Thinking mathematically.

Problem
Primary curriculum
Secondary curriculum
### Two Primes Make One Square

Can you make square numbers by adding two prime numbers together?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Third Dimension

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Corners

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factor Lines

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shopping Basket

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multiplication Squares

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Reach 100

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All the Digits

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Money Bags

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

Age 5 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
### Mystery Matrix

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Moons of Vuvv

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shapes on the Playground

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

Age 7 to 11

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
### Numerically Equal

Can you draw a square in which the perimeter is numerically equal to the area?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 1 Step 2 Step

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Forgot the Numbers

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Wonky Watches

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Egyptian Rope

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Many Times?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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
### Quadrilaterals

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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
### American Billions

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Daisy

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two and Two

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Nine Colours

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Where Can We Visit?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's Possible?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Neighbours

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Age 11 to 14

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
### Buying a Balloon

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A City of Towers

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 2, 4, 6, 8

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mixed-up Socks

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 6 Beads

If you put three beads onto a tens/ones abacus you can make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Dice

Find all the numbers that can be made by adding the dots on two dice.

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Five Coins

Ben has five coins in his pocket. How much money might he have?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Let's Investigate Triangles

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Cuboids

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Age 7 to 11

Challenge Level

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Roll These Dice

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plants

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

Age 5 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Age 7 to 14

Challenge Level