# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof ### Children at Large

##### Age 11 to 14Challenge Level

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? ### Cross-country Race

##### Age 14 to 16Challenge Level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places? ### Marbles

##### Age 11 to 14Challenge Level

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades? ### Even So

##### Age 11 to 14Challenge Level

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why? ### Take Three from Five

##### Age 11 to 16Challenge Level

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? ### Postage

##### Age 14 to 16Challenge Level

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . . ### A Biggy

##### Age 14 to 16Challenge Level

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power. ### Mod 3

##### Age 14 to 16Challenge Level

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3. ### 1 Step 2 Step

##### Age 11 to 14Challenge Level

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? ### Greetings

##### Age 11 to 14Challenge Level

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . . ### Tri-colour

##### Age 11 to 14Challenge Level

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs? ### N000ughty Thoughts

##### Age 14 to 16Challenge Level

How many noughts are at the end of these giant numbers? ### Elevenses

##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge Level

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour? ##### Age 14 to 16Challenge Level

Four jewellers share their stock. Can you work out the relative values of their gems? ### Doodles

##### Age 14 to 16Challenge Level

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections? ### Russian Cubes

##### Age 14 to 16Challenge Level

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that? ### Ordered Sums

##### Age 14 to 16Challenge Level

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . . ### What Numbers Can We Make Now?

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . . ### What Numbers Can We Make?

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

I am exactly n times my daughter's age. In m years I shall be ... How old am I? ### DOTS Division

##### Age 14 to 16Challenge Level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}. ### Square Mean

##### Age 14 to 16Challenge Level

Is the mean of the squares of two numbers greater than, or less than, the square of their means? ### Cosines Rule

##### Age 14 to 16Challenge Level

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement. ### Janine's Conjecture

##### Age 14 to 16Challenge Level

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . . ### Sixational

##### Age 14 to 18Challenge Level

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . . ### Ratty

##### Age 11 to 14Challenge Level

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? ### Disappearing Square

##### Age 11 to 14Challenge Level

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . ### Classifying Solids Using Angle Deficiency

##### Age 11 to 16Challenge Level

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry ### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities. ### The Triangle Game

##### Age 11 to 16Challenge Level

Can you discover whether this is a fair game? ### Top-heavy Pyramids

##### Age 11 to 14Challenge Level

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200. ### Mouhefanggai

##### Age 14 to 16

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. ### Seven Squares - Group-worthy Task

##### Age 11 to 14Challenge Level

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? ### Triangular Intersection

##### Age 14 to 16 ShortChallenge Level

What is the largest number of intersection points that a triangle and a quadrilateral can have? ### More Number Sandwiches

##### Age 11 to 16Challenge Level

When is it impossible to make number sandwiches? ### Gabriel's Problem

##### Age 11 to 14Challenge Level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was? ### Advent Calendar 2011 - Secondary

##### Age 11 to 18Challenge Level

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. ### Sticky Numbers

##### Age 11 to 14Challenge Level

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? ##### Age 11 to 14Challenge Level

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. ### The Great Weights Puzzle

##### Age 14 to 16Challenge Level

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest? ### Multiplication Square

##### Age 14 to 16Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? ### Some Circuits in Graph or Network Theory

##### Age 14 to 18

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits. ### Pattern of Islands

##### Age 11 to 14Challenge Level

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island... ### Sprouts Explained

##### Age 7 to 18

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . . ### Con Tricks

##### Age 11 to 14

Here are some examples of 'cons', and see if you can figure out where the trick is. ### Tourism

##### Age 11 to 14Challenge Level

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. ### Go Forth and Generalise

##### Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. ### Logic

##### Age 7 to 14

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.