Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

1 Step 2 Step

Age 11 to 14
Challenge 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?

Pattern of Islands

Age 11 to 14
Challenge Level

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

Children at Large

Age 11 to 14
Challenge 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 16
Challenge 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?

9 Weights

Age 11 to 14
Challenge 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?

The Great Weights Puzzle

Age 14 to 16
Challenge 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?

Greetings

Age 11 to 14
Challenge 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. . . .

Postage

Age 14 to 16
Challenge 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. . . .

Elevenses

Age 11 to 14
Challenge 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?

Sticky Numbers

Age 11 to 14
Challenge Level

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Flight of the Flibbins

Age 11 to 14
Challenge 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. . . .

Tri-colour

Age 11 to 14
Challenge 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?

Clocked

Age 11 to 14
Challenge 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?

Not Necessarily in That Order

Age 11 to 14
Challenge Level

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

Binomial Coefficients

Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

Knight Defeated

Age 14 to 16
Challenge 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. . . .

Master Minding

Age 11 to 14
Challenge Level

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

Russian Cubes

Age 14 to 16
Challenge Level

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

Doodles

Age 14 to 16
Challenge Level

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

Dicing with Numbers

Age 11 to 14
Challenge 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?

Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18
Challenge 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.

Calendar Capers

Age 11 to 14
Challenge Level

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

Largest Product

Age 11 to 14
Challenge Level

Which set of numbers that add to 10 have the largest product?

Top-heavy Pyramids

Age 11 to 14
Challenge Level

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

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.

The Genie in the Jar

Age 11 to 14
Challenge 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. . . .

More Mathematical Mysteries

Age 11 to 14
Challenge 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. . . .

Thirty Nine, Seventy Five

Age 11 to 14
Challenge Level

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

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. . . .

Geometry and Gravity 2

Age 11 to 18

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

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.

More Number Pyramids

Age 11 to 14
Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Eleven

Age 11 to 14
Challenge Level

Replace each letter with a digit to make this addition correct.

Adding All Nine

Age 11 to 14
Challenge Level

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Con Tricks

Age 11 to 14

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

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.

One O Five

Age 11 to 14
Challenge 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. . . .

Tis Unique

Age 11 to 14
Challenge Level

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

Gabriel's Problem

Age 11 to 14
Challenge Level

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

N000ughty Thoughts

Age 14 to 16
Challenge Level

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

Road Maker

Age 14 to 18
Challenge Level

Which of these roads will satisfy a Munchkin builder?

Advent Calendar 2011 - Secondary

Age 11 to 18
Challenge Level

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

More Marbles

Age 11 to 14
Challenge 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?

Marbles

Age 11 to 14
Challenge 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?

Take Three from Five

Age 11 to 16
Challenge 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?

How Many Dice?

Age 11 to 14
Challenge 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. . . .

Classifying Solids Using Angle Deficiency

Age 11 to 16
Challenge Level

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

Seven Squares - Group-worthy Task

Age 11 to 14
Challenge 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?

AMGM

Age 14 to 16
Challenge Level

Can you use the diagram to prove the AM-GM inequality?

Tower of Hanoi

Age 11 to 14
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.