Resources tagged with Mathematical reasoning & proof similar to To Prove or Not to Prove:
Other tags that relate to To Prove or Not to Prove
Visualising. Quadratic equations. Dynamical systems. Manipulating algebraic expressions/formulae. Generalising. Number theory. Patterned numbers. Making and proving conjectures. Inequality/inequalities. Mathematical reasoning & proof.There are 164 results
To Prove or Not to Prove
Stage: 5
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Sperner's Lemma
Stage: 5
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.
Iffy Logic
Stage: 4 Short Challenge Level: 
Can you rearrange the cards to make a series of correct mathematical statements?
Direct Logic
Stage: 5 Challenge Level:
Can you work through these direct proofs, using our interactive proof sorters?
Tree Graphs
Stage: 4 Challenge Level:
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree. . . .
The Great Weights Puzzle
Stage: 4 Challenge Level:
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
Natural Sum
Stage: 4 Challenge Level:
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
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.
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: 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.
Water Pistols
Stage: 5 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?
Diverging
Stage: 5 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
Stage: 4 and 5
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Russian Cubes
Stage: 4 Challenge Level:
How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . .
Binomial
Stage: 5 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
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.
Sprouts
Stage: 2, 3, 4 and 5
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. . . .
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?
Tetra Inequalities
Stage: 5 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.
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.
Gift of Gems
Stage: 4 Challenge Level:
Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .
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...
Rotating Triangle
Stage: 3 and 4 Challenge Level:
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Proof: A Brief Historical Survey
Stage: 4 and 5
If you think that mathematical proof is really clearcut and universal then you should read this article.
Picture Story
Stage: 4 Challenge Level:
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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.
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.
Long Short
Stage: 4 Challenge Level:
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
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. . . .
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.
Always Perfect
Stage: 4 Challenge Level:
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
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.
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?
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. . . .
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.
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?
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.
Big, Bigger, Biggest
Stage: 5 Challenge Level:
Which is the biggest and which the smallest of these numbers and by how do they compare in magnitude? A = 2000^{2002} B = 2001^{2001} C = 2002^{2000}
Proximity
Stage: 4 Challenge Level:
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
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?
Pair Squares
Stage: 5 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.
Janine's Conjecture
Stage: 4 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. . . .
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. . . .
Cyclic Quad Jigsaw
Stage: 4 Challenge Level:
A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
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