Other tags that relate to Cocked Hat
Symmetry. Expanding and factorising quadratics. Mathematical reasoning & proof. Graphing software and graphical calculators. Quadratic equations. Maths Supporting SET. Biology. Turning points. Graph sketching. Functions and their inverses.There are 173 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
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?
Fractional Calculus III
Age 16 to 18
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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.
Truth Tables and Electronic Circuits
Age 11 to 18
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
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.
Iff
Age 14 to 18
Challenge Level
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
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?
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.
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.
Big, Bigger, Biggest
Age 16 to 18
Challenge Level
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
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.
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?
There's a Limit
Age 14 to 18
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?
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/...)).
The Frieze Tree
Age 11 to 16
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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.
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.
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.
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. . . .
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?
Why 24?
Age 14 to 16
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.
Power Quady
Age 16 to 18
Challenge Level
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
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.
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.
Always Perfect
Age 14 to 18
Challenge Level
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Flexi Quad Tan
Age 16 to 18
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.
Common Divisor
Age 14 to 16
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.
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.
Problem Solving, Using and Applying and Functional Mathematics
Age 5 to 18
Challenge Level
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
Classifying Solids Using Angle Deficiency
Age 11 to 16
Challenge Level
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
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.
Mind Your Ps and Qs
Age 16 to 18 Short
Challenge Level
Sort these mathematical propositions into a series of 8 correct statements.
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?
Geometric Parabola
Age 14 to 16
Challenge Level
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
Areas and Ratios
Age 16 to 18
Challenge Level
Do you have enough information to work out the area of the shaded quadrilateral?
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?
Breaking the Equation ' Empirical Argument = Proof '
Age 7 to 18
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.
Notty Logic
Age 16 to 18
Challenge Level
Have a go at being mathematically negative, by negating these statements.
Exponential Intersection
Age 16 to 18
Challenge Level
Can the pdfs and cdfs of an exponential distribution intersect?
Iffy Logic
Age 14 to 18
Challenge Level
Can you rearrange the cards to make a series of correct mathematical statements?
Mouhefanggai
Age 14 to 16
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.
Air Nets
Age 7 to 18
Challenge Level
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Plus or Minus
Age 16 to 18
Challenge Level
Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
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.
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.