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Broad Topics > Using, Applying and Reasoning about Mathematics > Making and proving conjectures

Polycircles

Stage: 4 Challenge Level:

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Pentagon

Stage: 4 Challenge Level:

Find the vertices of a pentagon given the midpoints of its sides.

Triangles Within Triangles

Stage: 4 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

Triangles Within Pentagons

Stage: 4 Challenge Level:

Show that all pentagonal numbers are one third of a triangular number.

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?

Triangles Within Squares

Stage: 4 Challenge Level:

Can you find a rule which relates triangular numbers to square numbers?

Tri-split

Stage: 4 Challenge Level:

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

Multiplication Square

Stage: 3 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 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.

Dice, Routes and Pathways

Stage: 1, 2 and 3

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

Stage: 4 Challenge Level:

The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?

Always a Multiple?

Stage: 3 Challenge Level:

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Multiplication Arithmagons

Stage: 4 Challenge Level:

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Center Path

Stage: 3 and 4 Challenge Level:

Four rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of. . . .

What's Possible?

Stage: 4 Challenge Level:

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Pericut

Stage: 4 and 5 Challenge Level:

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

To Prove or Not to Prove

Stage: 4 and 5

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

Stage: 4 Challenge Level:

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

An Introduction to Magic Squares

Stage: 4 Challenge Level:

Points D, E and F are on the the sides of triangle ABC. Circumcircles are drawn to the triangles ADE, BEF and CFD respectively. What do you notice about these three circumcircles?

Loopy

Stage: 4 Challenge Level:

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Stage: 4 Challenge Level:

Explore the relationship between quadratic functions and their graphs.

Few and Far Between?

Stage: 4 and 5 Challenge Level:

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

Close to Triangular

Stage: 4 Challenge Level:

Drawing a triangle is not always as easy as you might think!

Curvy Areas

Stage: 4 Challenge Level:

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

A Little Light Thinking

Stage: 4 Challenge Level:

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Charlie's Mapping

Stage: 3 Challenge Level:

Charlie has created a mapping. Can you figure out what it does? What questions does it prompt you to ask?

Alison's Mapping

Stage: 4 Challenge Level:

Alison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

Consecutive Negative Numbers

Stage: 3 Challenge Level:

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Happy Numbers

Stage: 3 Challenge Level:

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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. . . .

DOTS Division

Stage: 4 Challenge Level:

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

Epidemic Modelling

Stage: 4 and 5 Challenge Level:

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

How Old Am I?

Stage: 4 Challenge Level:

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Helen's Conjecture

Stage: 3 Challenge Level:

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Exploring Simple Mappings

Stage: 3 Challenge Level:

Explore the relationship between simple linear functions and their graphs.
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