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Resources tagged with Mathematical reasoning & proof similar to How Many Solutions?:

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

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How Many Solutions?

Age 16 to 18 Challenge Level:

Find all the solutions to the this equation.

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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?

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Interpolating Polynomials

Age 16 to 18 Challenge Level:

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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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. . . .

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Power Quady

Age 16 to 18 Challenge Level:

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

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Sums of Squares and Sums of Cubes

Age 16 to 18

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.

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Look Before You Leap

Age 16 to 18 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

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AMGM

Age 14 to 16 Challenge Level:

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

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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. . . .

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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.

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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.

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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?

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More Sums of Squares

Age 16 to 18

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

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Binomial

Age 16 to 18 Challenge Level:

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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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.

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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?

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Where Do We Get Our Feet Wet?

Age 16 to 18

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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Telescoping Functions

Age 16 to 18

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.

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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!

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Three Ways

Age 16 to 18 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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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.

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Big, Bigger, Biggest

Age 16 to 18 Challenge Level:

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

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Sixational

Age 14 to 18 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. . . .

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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.

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Quadratic Harmony

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.

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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.

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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.

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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.

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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.

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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?

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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.

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Diophantine N-tuples

Age 14 to 16 Challenge Level:

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

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Pythagoras Proofs

Age 14 to 16 Challenge Level:

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

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Advent Calendar 2011 - Secondary

Age 11 to 18 Challenge Level:

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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Iffy Logic

Age 14 to 18 Challenge Level:

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

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Can it Be

Age 16 to 18 Challenge Level:

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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More Dicey Decisions

Age 16 to 18 Challenge Level:

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

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Modulus Arithmetic and a Solution to Differences

Age 16 to 18

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

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Magic Squares II

Age 14 to 18

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

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Picturing Pythagorean Triples

Age 14 to 18

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.

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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. . . .

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A Knight's Journey

Age 14 to 18

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

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Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.

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Pythagorean Triples II

Age 11 to 16

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

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Dodgy Proofs

Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?

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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.

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Little and Large

Age 16 to 18 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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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?

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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?

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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.