# Being Thoughtful

Being Thoughtful is part of our Developing Mathematical Habits of Mind collection.

Good thinkers are thoughtful and reflective. They think critically and can explain and justify the choices that they make. They become absorbed in what they are doing and show attention to detail. They look back on what they have done in order to learn from both successes and failures.

Want to become better at thinking mathematically?

These problems may require careful consideration. Immerse yourself in them - we think they are worth the effort!

You can browse through the Number, Algebra, Geometry or Statistics collections, or scroll down to see the full set of problems below.

### Being Thoughtful - Number

Number problems that require careful consideration.

### Being Thoughtful - Algebra

Algebra problems that require careful consideration.

### Being Thoughtful - Geometry

Geometry problems that require careful consideration.

### Being Thoughtful - Statistics

Statistics problems that require careful consideration.

### Dozens

##### KS 3 Challenge Level:

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

### How Old Am I?

##### KS 4 Challenge Level:

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?

### Funny Factorisation

##### KS 3 Challenge Level:

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can you find the factors?

### Where Can We Visit?

##### KS 3 Challenge Level:

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?

### Two and Two

##### KS 3 Challenge Level:

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

### Marbles in a Box

##### KS 3 Challenge Level:

How many winning lines can you make in a three-dimensional version of noughts and crosses?

### Attractive Tablecloths

##### KS 4 Challenge Level:

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

### In a Box

##### KS 4 Challenge Level:

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

### Triangles to Tetrahedra

##### KS 3 Challenge Level:

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

### Frogs

##### KS 3 Challenge Level:

How many moves does it take to swap over some red and blue frogs? Do you have a method?

##### KS 3 Challenge Level:

How many different symmetrical shapes can you make by shading triangles or squares?

### Hexy-metry

##### KS 4 Challenge Level:

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

### Special Numbers

##### KS 3 Challenge Level:

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

### The Spider and the Fly

##### KS 4 Challenge Level:

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

### Cuboids

##### KS 3 Challenge Level:

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

### On the Edge

##### KS 3 Challenge Level:

If you move the tiles around, can you make squares with different coloured edges?

### Square It

##### KS 3 & 4 Challenge Level:

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

### Isosceles Triangles

##### KS 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### Stars

##### KS 3 Challenge Level:

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

### Arithmagons

##### KS 4 Challenge Level:

Can you find the values at the vertices when you know the values on the edges?

##### KS 3 Challenge Level:

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and two properties of the shape.

### Shapely Pairs

##### KS 3 Challenge Level:

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

### Property Chart

##### KS 3 Challenge Level:

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

### Odds and Evens

##### KS 3 Challenge Level:

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

### Two's Company

##### KS 3 Challenge Level:

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

### Cosy Corner

##### KS 3 Challenge Level:

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

### Semi-regular Tessellations

##### KS 3 Challenge Level:

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

### Product Sudoku

##### KS 3 Challenge Level:

The clues for this Sudoku are the product of the numbers in adjacent squares.

### Substitution Cipher

##### KS 3 Challenge Level:

Find the frequency distribution for ordinary English, and use it to help you crack the code.

### Transformation Game

##### KS 3 Challenge Level:

Why not challenge a friend to play this transformation game?

### A Chance to Win?

##### KS 3 Challenge Level:

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

### Difference Sudoku

##### KS 4 Challenge Level:

Use the differences to find the solution to this Sudoku.

### Which Spinners?

##### KS 4 Challenge Level:

Can you work out which spinners were used to generate the frequency charts?

### M, M and M

##### KS 3 Challenge Level:

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

### Cops and Robbers

##### KS 2, 3 & 4 Challenge Level:

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

### The Remainders Game

##### KS 2 & 3 Challenge Level:

A game that tests your understanding of remainders.

### Nice or Nasty

##### KS 2 & 3 Challenge Level:

There are nasty versions of this dice game but we'll start with the nice ones...

### Dicey Operations

##### KS 3 Challenge Level:

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

### How Much Can We Spend?

##### KS 3 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

### Tower of Hanoi

##### KS 3 Challenge Level:

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

### Shifting Times Tables

##### KS 3 Challenge Level:

Can you find a way to identify times tables after they have been shifted up?

### A Little Light Thinking

##### KS 4 Challenge Level:

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

### Charlie's Delightful Machine

##### KS 3 & 4 Challenge Level:

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

### Sociable Cards

##### KS 3 Challenge Level:

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

### Last One Standing

##### KS 4 Challenge Level:

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

### What Does Random Look Like?

##### KS 3 Challenge Level:

Engage in a little mathematical detective work to see if you can spot the fakes.

### Which Solids Can We Make?

##### KS 3 Challenge Level:

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

### Missing Multipliers

##### KS 3 Challenge Level:

What is the smallest number of answers you need to reveal in order to work out the missing headers?

### Multiplication Arithmagons

##### KS 4 Challenge Level:

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

### Finding Factors

##### KS 4 Challenge Level:

Can you find the hidden factors which multiply together to produce each quadratic expression?

### Constructing Triangles

##### KS 3 Challenge Level:

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

### Perception Versus Reality

##### KS 4 & 5 Challenge Level:

Infographics are a powerful way of communicating statistical information. Can you come up with your own?

### Wipeout

##### KS 3 Challenge Level:

Can you do a little mathematical detective work to figure out which number has been wiped out?

### Box Plot Match

##### KS 4 Challenge Level:

Match the cumulative frequency curves with their corresponding box plots.

### Cryptarithms

##### KS 3 Challenge Level:

Can you crack these cryptarithms?

### Forwards Add Backwards

##### KS 3 Challenge Level:

What happens when you add a three digit number to its reverse?

### Reversals

##### KS 3 Challenge Level:

Where should you start, if you want to finish back where you started?

### Unequal Averages

##### KS 3 Challenge Level:

Play around with sets of five numbers and see what you can discover about different types of average...

### Gabriel's Problem

##### KS 3 Challenge Level:

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?