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#### Resources tagged with Making and proving conjectures similar to Wrapping Gifts:

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### There are 57 results

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

### Triangles Within Pentagons

##### Stage: 4 Challenge Level:

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

### Triangles Within Triangles

##### Stage: 4 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

### 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.

### Triangles Within Squares

##### Stage: 4 Challenge Level:

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

### Cyclic Triangles

##### Stage: 5 Challenge Level:

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

### Thebault's Theorem

##### Stage: 5 Challenge Level:

Take any parallelogram and draw squares on the sides of the parallelogram. What can you prove about the quadrilateral formed by joining the centres of these squares?

### 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?

### Vecten

##### Stage: 5 Challenge Level:

Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.

### 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.

### Trig Rules OK

##### Stage: 5 Challenge Level:

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

### Pythagorean Fibs

##### Stage: 5 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

### Discrete Trends

##### Stage: 5 Challenge Level:

Find the maximum value of n to the power 1/n and prove that it is a maximum.

### Fibonacci Fashion

##### Stage: 5 Challenge Level:

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

### 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?

### 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?

### Painting by Numbers

##### Stage: 5 Challenge Level:

How many different colours of paint would be needed to paint these pictures by numbers?

### 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?

### 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?

### 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!

##### Stage: 4 Challenge Level:

Explore the relationship between quadratic functions and their graphs.

### Fixing It

##### Stage: 5 Challenge Level:

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

### Prime Sequences

##### Stage: 5 Challenge Level:

This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?

### 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?

### Steve's Mapping

##### Stage: 5 Challenge Level:

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

### 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. . . .

### Conjugate Tracker

##### Stage: 5 Challenge Level:

Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.

### OK! Now Prove It

##### Stage: 5 Challenge Level:

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

### Least of All

##### Stage: 5 Challenge Level:

A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.

##### Stage: 5 Challenge Level:

With red and blue beads on a circular wire; 'put a red bead between any two of the same colour and a blue between different colours then remove the original beads'. Keep repeating this. What happens?

##### 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?

### 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?

### Integral Sandwich

##### Stage: 5 Challenge Level:

Generalise this inequality involving integrals.

### 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?

### Sixty-seven Squared

##### Stage: 5 Challenge Level:

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

##### 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?

### 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?

### Why Stop at Three by One

##### Stage: 5

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

### The Kth Sum of N Numbers

##### Stage: 5

Yatir from Israel describes his method for summing a series of triangle numbers.

### 2^n -n Numbers

##### Stage: 5

Yatir from Israel wrote this article on numbers that can be written as $2^n-n$ where n is a positive integer.

### 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?

### Binary Squares

##### Stage: 5 Challenge Level:

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

### Poly Fibs

##### Stage: 5 Challenge Level:

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

### Summats Clear

##### Stage: 5 Challenge Level:

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

### Fibonacci Factors

##### Stage: 5 Challenge Level:

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

### 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?

### 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.

### 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}.