Resources tagged with Mathematical reasoning & proof similar to Which Twin Is Older?:
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Vectors. Matrices. Time. Relative velocity. Mathematical reasoning & proof. Vector notation. History of measurement. Scalar products. Addition of vectors. Vector algebra.There are 164 results
Flexi Quad Tan
Stage: 5 Challenge Level: 
As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.
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.
The Golden Ratio, Fibonacci Numbers and Continued Fractions.
Stage: 4
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
Stone Henge
Stage: 5 Challenge Level:
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Magic Squares II
Stage: 4 and 5
An article which gives an account of some properties of magic squares.
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.
Recent Developments on S.P. Numbers
Stage: 5
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
Try to Win
Stage: 5
Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
Where Do We Get Our Feet Wet?
Stage: 5
Jenny Piggott chose this article. Professor Körner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Modulus Arithmetic and a Solution to Differences
Stage: 5
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
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. . . .
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 orf two or more cubes.
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.
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.
Classifying Solids Using Angle Deficiency
Stage: 3 and 4
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Continued Fractions II
Stage: 5
In this article we show that every whole number can be written as a continued fraction of the form k/[1+k/[1+k/etc.]].
The Frieze Tree
Stage: 4
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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.
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.
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.
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?
Power Quady
Stage: 4 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Modular Fractions
Stage: 5 Challenge Level:
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
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.
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. . . .
Magic W Wrap Up
Stage: 5 Challenge Level:
Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.
Logic, Truth Tables and Switching Circuits Challenge
Stage: 2, 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. . . .
Mechanical Integration
Stage: 5 Challenge Level:
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Quadratic Harmony
Stage: 5 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.
Why 24?
Stage: 4 Challenge Level:
Take any prime number greater than 3, square it, subtract one and divide by 24. Make a statement about what you notice about dividing by 24 (a conjecture) and prove that what you say is always true.
Little and Large
Stage: 5 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?
Whole Number Dynamics II
Stage: 3, 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.
Whole Number Dynamics III
Stage: 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 I
Stage: 3, 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.
Can it Be
Stage: 5 Challenge Level:
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Pythagorean Triples I
Stage: 4 and 5
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!
Pythagorean Triples II
Stage: 4 and 5
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
Euler's Formula and Topology
Stage: 4 and 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. . . .
Salinon
Stage: 4 Challenge Level:
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?
Binary Sequences
Stage: 5 Challenge Level:
Show that the infinite set of finite (or terminating) binary sequences can be written as an ordered list whereas the infinite set of all infinite binary sequences cannot.
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}.
Symmetric Tangles
Stage: 4
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
And So on - and on -and On
Stage: 5 Challenge Level:
Can you find the value of this function involving algebraic fractions for x=2000?
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.
Cube Net
Stage: 5 Challenge Level:
How many tours visit each vertex of a cube once and only once? How many return to the starting point?
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