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Resources tagged with Mathematical reasoning & proof similar to Napier's Location Arithmetic:

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

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Number Rules - OK

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

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...

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Composite Notions

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

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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Proof Sorter - Quadratic Equation

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

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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For What?

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

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

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To Prove or Not to Prove

Stage: 4 and 5

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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Chocolate Maths

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Archimedes and Numerical Roots

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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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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One O Five

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

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. . . .

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A Biggy

Stage: 4 Challenge Level: Challenge Level:1

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.

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Power Mad!

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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What Numbers Can We Make Now?

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

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

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Multiplication Square

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

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

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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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Never Prime

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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Sixational

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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Mod 3

Stage: 4 Challenge Level: Challenge Level:1

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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DOTS Division

Stage: 4 Challenge Level: Challenge Level:1

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}.

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Converse

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

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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Diophantine N-tuples

Stage: 4 Challenge Level: Challenge Level:1

Can you explain why a sequence of operations always gives you perfect squares?

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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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Common Divisor

Stage: 4 Challenge Level: Challenge Level:1

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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Long Short

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

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

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Always Perfect

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

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

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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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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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Pythagorean Triples I

Stage: 3 and 4

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!

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Take Three from Five

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

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?

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Angle Trisection

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

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

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Postage

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Road Maker

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

Which of these roads will satisfy a Munchkin builder?

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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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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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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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Tessellating Hexagons

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

Which hexagons tessellate?

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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.

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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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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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Seven Squares - Group-worthy Task

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

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?

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There's a Limit

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

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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Our Ages

Stage: 4 Challenge Level: Challenge Level:1

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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Square Mean

Stage: 4 Challenge Level: Challenge Level:1

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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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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Salinon

Stage: 4 Challenge Level: Challenge Level:1

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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Pythagorean Triples II

Stage: 3 and 4

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

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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.

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Perfectly Square

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

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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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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Three Frogs

Stage: 4 Challenge Level: Challenge Level:1

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

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Mediant

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

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.