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Resources tagged with Mathematical reasoning & proof similar to Square Product:

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

Common Divisor

Stage: 4 Challenge Level:

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

For What?

Stage: 4 Challenge Level:

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

Why 24?

Stage: 4 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

A Biggy

Stage: 4 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.

Mod 3

Stage: 4 Challenge Level:

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

Sixational

Stage: 4 and 5 Challenge Level:

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. . . .

Number Rules - OK

Stage: 4 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...

Take Three from Five

Stage: 4 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?

DOTS Division

Stage: 4 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}.

Symmetric Tangles

Stage: 4

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

Cosines Rule

Stage: 4 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.

N000ughty Thoughts

Stage: 4 Challenge Level:

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

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. . . .

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.

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.

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.

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?

Magic Squares II

Stage: 4 and 5

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

A Knight's Journey

Stage: 4 and 5

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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.

Mouhefanggai

Stage: 4

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.

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. . . .

The Triangle Game

Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?

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

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.

The Frieze Tree

Stage: 3 and 4

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Pythagorean Triples II

Stage: 3 and 4

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

Always Perfect

Stage: 4 Challenge Level:

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

Logic, Truth Tables and Switching Circuits Challenge

Stage: 3, 4 and 5

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

Fitting In

Stage: 4 Challenge Level:

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

Square Mean

Stage: 4 Challenge Level:

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

Long Short

Stage: 4 Challenge Level:

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

Postage

Stage: 4 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. . . .

Mediant

Stage: 4 Challenge Level:

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.

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!

There's a Limit

Stage: 4 and 5 Challenge Level:

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?

Stage: 4 Challenge Level:

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

Triangle Incircle Iteration

Stage: 4 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

Angle Trisection

Stage: 4 Challenge Level:

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

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.

Similarly So

Stage: 4 Challenge Level:

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Pythagoras Proofs

Stage: 4 Challenge Level:

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

Breaking the Equation ' Empirical Argument = Proof '

Stage: 2, 3, 4 and 5

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

Areas and Ratios

Stage: 4 Challenge Level:

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Folding Fractions

Stage: 4 Challenge Level:

What fractions can you divide the diagonal of a square into by simple folding?

Stage: 3 Challenge Level:

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

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.

Iffy Logic

Stage: 4 and 5 Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements?

Elevenses

Stage: 3 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?

Geometric Parabola

Stage: 4 Challenge Level:

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.