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Resources tagged with Mathematical reasoning & proof similar to Last Biscuit:

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

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?

Cross-country Race

Age 11 to 14 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?

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. . . .

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?

Age 14 to 16 Challenge Level:

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

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?

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. . . .

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. . . .

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. . . .

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?

Con Tricks

Age 11 to 14

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

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?

N000ughty Thoughts

Age 14 to 16 Challenge Level:

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

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. . . .

Age 14 to 16 Challenge Level:

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

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?

Proof: A Brief Historical Survey

Age 14 to 18

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

Mouhefanggai

Age 14 to 16

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

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?

Cycle It

Age 11 to 14 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.

Leonardo's Problem

Age 14 to 18 Challenge Level:

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?

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. . . .

For What?

Age 14 to 16 Challenge Level:

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Towering Trapeziums

Age 14 to 16 Challenge Level:

Can you find the areas of the trapezia in this sequence?

Dalmatians

Age 14 to 18 Challenge Level:

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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. . . .

Number Rules - OK

Age 14 to 16 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Matter of Scale

Age 14 to 16 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

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?

Symmetric Tangles

Age 14 to 16

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

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. . . .

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. . . .

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.

Always Perfect

Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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. . . .

Impossible Sandwiches

Age 11 to 18

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.

Pythagorean Triples II

Age 11 to 16

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Pythagorean Triples I

Age 11 to 16

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

Unit Interval

Age 14 to 18 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?

Take Three from Five

Age 14 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?

Angle Trisection

Age 14 to 16 Challenge Level:

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Ratty

Age 11 to 14 Challenge Level:

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

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.

Cosines Rule

Age 14 to 16 Challenge Level:

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

A Biggy

Age 14 to 16 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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 Triangle Game

Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game?

Long Short

Age 14 to 16 Challenge Level:

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Volume of a Pyramid and a Cone

Age 11 to 14

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.