# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof ### The Triangle Game

##### Age 11 to 16Challenge Level

Can you discover whether this is a fair game? ### Clocked

##### Age 11 to 14Challenge Level

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? ### 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? ### One O Five

##### Age 11 to 14Challenge Level

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . . ### 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 ### How Many Dice?

##### Age 11 to 14Challenge Level

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . . ### 9 Weights

##### Age 11 to 14Challenge Level

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance? ### 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? ### Yih or Luk Tsut K'i or Three Men's Morris

##### Age 11 to 18Challenge Level

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . ### Convex Polygons

##### Age 11 to 14Challenge Level

Show that among the interior angles of a convex polygon there cannot be more than three acute angles. ### Natural Sum

##### Age 14 to 16Challenge Level

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . . ### Picture Story

##### Age 14 to 16Challenge Level

Can you see how this picture illustrates the formula for the sum of the first six cube numbers? ### Proximity

##### Age 14 to 16Challenge Level

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours. ### Concrete Wheel

##### Age 11 to 14Challenge Level

A huge wheel is rolling past your window. What do you see? ### 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? ### Dicing with Numbers

##### Age 11 to 14Challenge Level

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal? ### Aba

##### Age 11 to 14Challenge Level

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct. ### KÃ¶nigsberg

##### Age 11 to 14Challenge Level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? ### 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. . . . ### The Genie in the Jar

##### Age 11 to 14Challenge Level

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . . ### Flight of the Flibbins

##### Age 11 to 14Challenge Level

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . . ### Calendar Capers

##### Age 11 to 14Challenge Level

Choose any three by three square of dates on a calendar page... ### Largest Product

##### Age 11 to 14Challenge Level

Which set of numbers that add to 10 have the largest product? ### More Mathematical Mysteries

##### Age 11 to 14Challenge Level

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . . ### 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. . . . ### Eleven

##### Age 11 to 14Challenge Level

Replace each letter with a digit to make this addition correct. ### 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. . . . ### Tower of Hanoi

##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge Level

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . . ### AMGM

##### Age 14 to 16Challenge Level

Can you use the diagram to prove the AM-GM inequality? ### More Number Pyramids

##### Age 11 to 14Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... ### 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. ### 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. ### 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. . . . ### 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? ### 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? ### A Long Time at the Till

##### Age 14 to 18Challenge Level

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? ### L-triominoes

##### Age 14 to 16Challenge Level

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way? ### 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? ### Problem Solving, Using and Applying and Functional Mathematics

##### Age 5 to 18Challenge Level

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. ### 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? ### Advent Calendar 2011 - Secondary

##### Age 11 to 18Challenge Level

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. ### 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}. ##### Age 14 to 16Challenge Level

Four jewellers share their stock. Can you work out the relative values of their gems? ##### Age 11 to 16Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem? ### Same Length

##### Age 11 to 16Challenge Level

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it? ### Calculating with Cosines

##### Age 14 to 18Challenge Level

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle? ### Tessellating Hexagons

##### Age 11 to 14Challenge Level

Which hexagons tessellate? ### 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? ### 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?