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Resources tagged with Mathematical reasoning & proof similar to Em'power'ed:

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

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.

Plus or Minus

Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Sums of Squares and Sums of Cubes

Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

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. . . .

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.

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

Ordered Sums

Stage: 4 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. . . .

Big, Bigger, Biggest

Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Dalmatians

Stage: 4 and 5 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.

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.

The Root Cause

Stage: 5 Challenge Level:

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

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}.

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.

Polite Numbers

Stage: 5 Challenge Level:

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Telescoping Functions

Stage: 5

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Three Frogs

Stage: 4 Challenge Level:

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. . . .

Golden Eggs

Stage: 5 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Stage: 3 and 4 Challenge Level:

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

On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

Perfectly Square

Stage: 4 Challenge Level:

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

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...

Archimedes and Numerical Roots

Stage: 4 Challenge Level:

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?

More Sums of Squares

Stage: 5

Tom writes about expressing numbers as the sums of three squares.

Modulus Arithmetic and a Solution to Dirisibly Yours

Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Can it Be

Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Rational Roots

Stage: 5 Challenge Level:

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Proofs with Pictures

Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Euler's Formula and Topology

Stage: 5

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

A Computer Program to Find Magic Squares

Stage: 5

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

The Triangle Game

Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?

Fractional Calculus III

Stage: 5

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Without Calculus

Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Euclid's Algorithm II

Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

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.

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.

Pareq Exists

Stage: 4 Challenge Level:

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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?

Circle Box

Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Rhombus in Rectangle

Stage: 4 Challenge Level:

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Composite Notions

Stage: 4 Challenge Level:

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

Stage: 5 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

Square Pair Circles

Stage: 5 Challenge Level:

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Euler's Squares

Stage: 4 Challenge Level:

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Transitivity

Stage: 5

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

Zig Zag

Stage: 4 Challenge Level:

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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?

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.