Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

The Triangle Game

Age 11 to 16
Challenge Level

Can you discover whether this is a fair game?

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. . . .

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?

Disappearing Square

Age 11 to 14
Challenge 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. . . .

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?

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?

More Number Pyramids

Age 11 to 14
Challenge Level

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

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?

Konigsberg Plus

Age 11 to 14
Challenge Level

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Eleven

Age 11 to 14
Challenge Level

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

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.

Chocolate Maths

Age 11 to 14
Challenge 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. . . .

Aba

Age 11 to 14
Challenge 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.

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. . . .

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. . . .

Picture Story

Age 14 to 16
Challenge Level

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

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?

Concrete Wheel

Age 11 to 14
Challenge Level

A huge wheel is rolling past your window. What do you see?

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?

Largest Product

Age 11 to 14
Challenge Level

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

Janine's Conjecture

Age 14 to 16
Challenge 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 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. . . .

Convex Polygons

Age 11 to 14
Challenge Level

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

Yih or Luk Tsut K'i or Three Men's Morris

Age 11 to 18
Challenge 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. . . .

Mediant Madness

Age 14 to 16
Challenge Level

Kyle and his teacher disagree about his test score - who is right?

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?

Königsberg

Age 11 to 14
Challenge 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?

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 14
Challenge 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.

Multiplication Square

Age 14 to 16
Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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?

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. . . .

What Numbers Can We Make Now?

Age 11 to 14
Challenge Level

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

Ordered Sums

Age 14 to 16
Challenge 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. . . .

Same Length

Age 11 to 16
Challenge Level

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

What Numbers Can We Make?

Age 11 to 14
Challenge Level

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

L-triominoes

Age 14 to 16
Challenge Level

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Cyclic Quadrilaterals

Age 11 to 16
Challenge Level

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

Pythagoras Proofs

Age 14 to 16
Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

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?

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. . . .

Tessellating Hexagons

Age 11 to 14
Challenge Level

Which hexagons tessellate?

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...

Natural Sum

Age 14 to 16
Challenge 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. . . .

Gift of Gems

Age 14 to 16
Challenge Level

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

DOTS Division

Age 14 to 16
Challenge 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}.

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.

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.