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Resources tagged with Mathematical reasoning & proof similar to Going First:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

Concrete Wheel

Stage: 3 Challenge Level:

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

Tourism

Stage: 3 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.

Königsberg

Stage: 3 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?

How Many Dice?

Stage: 3 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. . . .

Convex Polygons

Stage: 3 Challenge Level:

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

Tessellating Hexagons

Stage: 3 Challenge Level:

Which hexagons tessellate?

9 Weights

Stage: 3 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?

Flight of the Flibbins

Stage: 3 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. . . .

Online

Stage: 2 and 3 Challenge Level:

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Triangle Inequality

Stage: 3 Challenge Level:

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

Konigsberg Plus

Stage: 3 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.

Volume of a Pyramid and a Cone

Stage: 3

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Clocked

Stage: 3 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?

Sticky Numbers

Stage: 3 Challenge Level:

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

Go Forth and Generalise

Stage: 3

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.

Shuffle Shriek

Stage: 3 Challenge Level:

Can you find all the 4-ball shuffles?

One O Five

Stage: 3 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. . . .

More Number Pyramids

Stage: 3 Challenge Level:

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

The Triangle Game

Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?

Pattern of Islands

Stage: 3 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...

Disappearing Square

Stage: 3 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. . . .

Chocolate Maths

Stage: 3 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. . . .

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

Stage: 3, 4 and 5 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. . . .

Not Necessarily in That Order

Stage: 3 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?

Aba

Stage: 3 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.

Dicing with Numbers

Stage: 3 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?

Tis Unique

Stage: 3 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.

Calendar Capers

Stage: 3 Challenge Level:

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

Always the Same

Stage: 3 Challenge Level:

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 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.

Sprouts Explained

Stage: 2, 3, 4 and 5

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. . . .

Football Champs

Stage: 3 Challenge Level:

Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches.

Hockey

Stage: 3 Challenge Level:

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

Top-heavy Pyramids

Stage: 3 Challenge Level:

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

Classifying Solids Using Angle Deficiency

Stage: 3 and 4 Challenge Level:

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

Stage: 3 Challenge Level:

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Picture Story

Stage: 4 Challenge Level:

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

Eleven

Stage: 3 Challenge Level:

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

Pyramids

Stage: 3 Challenge Level:

What are the missing numbers in the pyramids?

Children at Large

Stage: 3 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?

Stage: 3 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?

Air Nets

Stage: 2, 3, 4 and 5 Challenge Level:

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Some Circuits in Graph or Network Theory

Stage: 4 and 5

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

Proof: A Brief Historical Survey

Stage: 4 and 5

If you think that mathematical proof is really clearcut and universal then you should read this article.

Natural Sum

Stage: 4 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. . . .

Rotating Triangle

Stage: 3 and 4 Challenge Level:

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Con Tricks

Stage: 3

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

Cycle It

Stage: 3 Challenge Level:

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Unit Interval

Stage: 4 and 5 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Magic Squares II

Stage: 4 and 5

An article which gives an account of some properties of magic squares.