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Resources tagged with Mathematical reasoning & proof similar to Equation Sudoku:

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

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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.

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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. . . .

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Tourism

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Picture Story

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Concrete Wheel

Stage: 3 Challenge Level: Challenge Level:1

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

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Shuffle Shriek

Stage: 3 Challenge Level: Challenge Level:1

Can you find all the 4-ball shuffles?

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How Many Dice?

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Advent Calendar 2011 - Secondary

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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

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Symmetric Tangles

Stage: 4

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

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Konigsberg Plus

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Online

Stage: 2 and 3 Challenge Level: Challenge Level:1

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.

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Sticky Numbers

Stage: 3 Challenge Level: Challenge Level:1

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

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The Great Weights Puzzle

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Not Necessarily in That Order

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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9 Weights

Stage: 3 Challenge Level: Challenge Level:1

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?

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Marbles

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Tri-colour

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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. . . .

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Hockey

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Knight Defeated

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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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.

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Pattern of Islands

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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The Triangle Game

Stage: 3 and 4 Challenge Level: Challenge Level:1

Can you discover whether this is a fair game?

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Königsberg

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Always the Same

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level: Challenge Level:1

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.

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Children at Large

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Natural Sum

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Disappearing Square

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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Gift of Gems

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Tessellating Hexagons

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Which hexagons tessellate?

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Cycle It

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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N000ughty Thoughts

Stage: 4 Challenge Level: Challenge Level:1

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

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Clocked

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Rotating Triangle

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Con Tricks

Stage: 3

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

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Leonardo's Problem

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

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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Ordered Sums

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Unit Interval

Stage: 4 and 5 Challenge Level: Challenge Level:1

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

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Whole Number Dynamics IV

Stage: 4 and 5

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

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Pent

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

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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Whole Number Dynamics III

Stage: 4 and 5

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

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Whole Number Dynamics II

Stage: 4 and 5

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

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Whole Number Dynamics V

Stage: 4 and 5

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

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Picturing Pythagorean Triples

Stage: 4 and 5

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

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Classifying Solids Using Angle Deficiency

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Impossible Sandwiches

Stage: 3, 4 and 5

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

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Paradoxes

Stage: 2 and 3

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

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Magic Squares II

Stage: 4 and 5

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

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Whole Number Dynamics I

Stage: 4 and 5

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.