Resources tagged with: Mathematical reasoning & proof

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

Rational Roots

Age 16 to 18 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.

The Clue Is in the Question

Age 16 to 18 Challenge Level:

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

Plus or Minus

Age 16 to 18 Challenge Level:

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

The Root Cause

Age 16 to 18 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.)

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?

Age 16 to 18 Short Challenge Level:

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

To Prove or Not to Prove

Age 14 to 18

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

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?

Impossible Triangles?

Age 16 to 18 Challenge Level:

Which of these triangular jigsaws are impossible to finish?

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.

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. . . .

Contrary Logic

Age 16 to 18 Challenge Level:

Can you invert the logic to prove these statements?

Age 14 to 18 Challenge Level:

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.

Continued Fractions II

Age 16 to 18

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Polynomial Relations

Age 16 to 18 Challenge Level:

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Iffy Logic

Age 14 to 18 Challenge Level:

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

Triangular Intersection

Age 14 to 16 Short Challenge Level:

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Thousand Words

Age 16 to 18 Challenge Level:

Here the diagram says it all. Can you find the diagram?

Square Mean

Age 14 to 16 Challenge Level:

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

Target Six

Age 16 to 18 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Pair Squares

Age 16 to 18 Challenge Level:

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Golden Eggs

Age 16 to 18 Challenge Level:

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

Mechanical Integration

Age 16 to 18 Challenge Level:

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Age 16 to 18 Challenge Level:

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

Pareq Exists

Age 14 to 16 Challenge Level:

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

Square Pair Circles

Age 16 to 18 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.

Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.

Magic Squares II

Age 14 to 18

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

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.

Sperner's Lemma

Age 16 to 18

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

Euclid's Algorithm II

Age 16 to 18

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.

Age 14 to 18 Challenge Level:

Which of these roads will satisfy a Munchkin builder?

On the Importance of Pedantry

Age 16 to 18

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

More Number Sandwiches

Age 11 to 16 Challenge Level:

When is it impossible to make number sandwiches?

Tetra Inequalities

Age 16 to 18 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

No Right Angle Here

Age 14 to 16 Challenge Level:

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

A Long Time at the Till

Age 14 to 18 Challenge Level:

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Dodgy Proofs

Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?

Calculating with Cosines

Age 14 to 18 Challenge Level:

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

Same Length

Age 11 to 16 Challenge Level:

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Perfectly Square

Age 14 to 16 Challenge Level:

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

Converse

Age 14 to 16 Challenge Level:

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?

Water Pistols

Age 16 to 18 Challenge Level:

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

Basic Rhythms

Age 16 to 18 Challenge Level:

Explore a number pattern which has the same symmetries in different bases.

Diverging

Age 16 to 18 Challenge Level:

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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.

AMGM

Age 14 to 16 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

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. . . .

Without Calculus

Age 16 to 18 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.