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Resources tagged with Mathematical reasoning & proof similar to Complex Countdown:

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

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The Triangle Game

Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game?

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Napoleon's Hat

Age 16 to 18 Challenge Level:

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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Stonehenge

Age 16 to 18 Challenge Level:

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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Cosines Rule

Age 14 to 16 Challenge Level:

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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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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Rolling Coins

Age 14 to 16 Challenge Level:

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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Proof Sorter - Geometric Series

Age 16 to 18 Challenge Level:

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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Proof Sorter - Quadratic Equation

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.

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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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Picture Story

Age 14 to 16 Challenge Level:

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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Natural Sum

Age 14 to 16 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. . . .

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Triangle Incircle Iteration

Age 14 to 16 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

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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.

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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.

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Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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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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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.

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Matter of Scale

Age 14 to 16 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

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Round and Round

Age 14 to 16 Challenge Level:

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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Circle Box

Age 14 to 16 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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Thousand Words

Age 16 to 18 Challenge Level:

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

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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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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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The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Age 14 to 16

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.

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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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The Great Weights Puzzle

Age 14 to 16 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?

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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.

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Direct Logic

Age 16 to 18 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

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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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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?

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Truth Tables and Electronic Circuits

Age 11 to 18

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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Proximity

Age 14 to 16 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.

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Middle Man

Age 16 to 18 Challenge Level:

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

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Shape and Territory

Age 16 to 18 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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Proof Sorter - Sum of an AP

Age 16 to 18 Challenge Level:

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

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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.

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Three Balls

Age 14 to 16 Challenge Level:

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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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?

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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.

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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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Rhombus in Rectangle

Age 14 to 16 Challenge Level:

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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Cube Net

Age 16 to 18 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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Generally Geometric

Age 16 to 18 Challenge Level:

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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Euler's Squares

Age 14 to 16 Challenge Level:

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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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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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.

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The Pillar of Chios

Age 14 to 16 Challenge Level:

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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Pythagorean Golden Means

Age 16 to 18 Challenge Level:

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

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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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Pent

Age 14 to 18 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.