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#### Resources tagged with Mathematical reasoning & proof similar to Cyclic Quadrilaterals:

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Challenge level:

### There are 177 results

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

### Cyclic Quadrilaterals

##### Stage: 3 Challenge Level:

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

### Tri-colour

##### Stage: 3 Challenge Level:

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

### The Genie in the Jar

##### Stage: 3 Challenge Level:

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

### Cross-country Race

##### Stage: 3 Challenge Level:

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

### Cycle It

##### Stage: 3 Challenge Level:

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

### Marbles

##### Stage: 3 Challenge Level:

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

### More Marbles

##### Stage: 3 Challenge Level:

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

### Ratty

##### Stage: 3 Challenge Level:

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

### Growing Ls

##### Stage: 3 Challenge Level:

Can you fit Ls together to make larger versions of themselves?

### Greetings

##### Stage: 3 Challenge Level:

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

### Winning Team

##### Stage: 3 Challenge Level:

Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?

### Thirty Nine, Seventy Five

##### Stage: 3 Challenge Level:

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

### Con Tricks

##### Stage: 3

Here are some examples of 'cons', and see if you can figure out where the trick is.

### More Mathematical Mysteries

##### Stage: 3 Challenge Level:

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

### Logic

##### Stage: 2 and 3

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

### Aba

##### Stage: 3 Challenge Level:

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

### Not Necessarily in That Order

##### Stage: 3 Challenge Level:

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

### Unit Fractions

##### Stage: 3 Challenge Level:

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

### Paradoxes

##### Stage: 2 and 3

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

### A Chordingly

##### Stage: 3 Challenge Level:

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

### Go Forth and Generalise

##### Stage: 3

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

### Tis Unique

##### Stage: 3 Challenge Level:

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

### Always the Same

##### Stage: 3 Challenge Level:

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

### AMGM

##### Stage: 4 Challenge Level:

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

### One O Five

##### Stage: 3 Challenge Level:

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

### Is it Magic or Is it Maths?

##### Stage: 3 Challenge Level:

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

### Hockey

##### Stage: 3 Challenge Level:

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

### Pyramids

##### Stage: 3 Challenge Level:

What are the missing numbers in the pyramids?

### 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?

### 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. . . .

### More Number Pyramids

##### Stage: 3 and 4 Challenge Level:

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

### Top-heavy Pyramids

##### Stage: 3 Challenge Level:

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

### Coins on a Plate

##### Stage: 3 Challenge Level:

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

### Pattern of Islands

##### Stage: 3 Challenge Level:

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

### 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. . . .

### Concrete Wheel

##### Stage: 3 Challenge Level:

A huge wheel is rolling past your window. What do you see?

### Mediant

##### Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

### 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. . . .

### Picture Story

##### Stage: 4 Challenge Level:

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

### Leonardo's Problem

##### Stage: 4 and 5 Challenge Level:

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

### 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. . . .

### Convex Polygons

##### Stage: 3 Challenge Level:

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

### Volume of a Pyramid and a Cone

##### Stage: 3

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

### The Pillar of Chios

##### Stage: 3 Challenge Level:

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

### Eleven

##### Stage: 3 Challenge Level:

Replace each letter with a digit to make this addition correct.

### Adding All Nine

##### Stage: 3 Challenge Level:

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

### 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.

### Even So

##### Stage: 3 Challenge Level:

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

### Mindreader

##### Stage: 3 Challenge Level:

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

### No Right Angle Here

##### Stage: 4 Challenge Level:

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
The text passage contains a series of mathematical problems and proofs, but no explicit mathematical questions or calculations are presented in a format that requires a specific numerical answer. The content is structured as a collection of logic and reasoning problems, with each problem or proof building on mathematical concepts and challenging the reader to think critically.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Search by Topic

#### Resources tagged with Mathematical reasoning & proof similar to Cyclic Quadrilaterals:

Filter by: Content type:
Stage:
Challenge level:

### There are 177 results

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

### Cyclic Quadrilaterals

##### Stage: 3 Challenge Level:

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

### Tri-colour

##### Stage: 3 Challenge Level:

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

### The Genie in the Jar

##### Stage: 3 Challenge Level:

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

### Cross-country Race

##### Stage: 3 Challenge Level:

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

### Cycle It

##### Stage: 3 Challenge Level:

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

### Marbles

##### Stage: 3 Challenge Level:

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

### More Marbles

##### Stage: 3 Challenge Level:

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

### Ratty

##### Stage: 3 Challenge Level:

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

### Growing Ls

##### Stage: 3 Challenge Level:

Can you fit Ls together to make larger versions of themselves?

### Greetings

##### Stage: 3 Challenge Level:

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

### Winning Team

##### Stage: 3 Challenge Level:

Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?

### Thirty Nine, Seventy Five

##### Stage: 3 Challenge Level:

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

### Con Tricks

##### Stage: 3

Here are some examples of 'cons', and see if you can figure out where the trick is.

### More Mathematical Mysteries

##### Stage: 3 Challenge Level:

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

### Logic

##### Stage: 2 and 3

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

### Aba

##### Stage: 3 Challenge Level:

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

### Not Necessarily in That Order

##### Stage: 3 Challenge Level:

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

### Unit Fractions

##### Stage: 3 Challenge Level:

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

### Paradoxes

##### Stage: 2 and 3

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

### A Chordingly

##### Stage: 3 Challenge Level:

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

### Go Forth and Generalise

##### Stage: 3

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

### Tis Unique

##### Stage: 3 Challenge Level:

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

### Always the Same

##### Stage: 3 Challenge Level:

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

### AMGM

##### Stage: 4 Challenge Level:

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

### One O Five

##### Stage: 3 Challenge Level:

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

### Is it Magic or Is it Maths?

##### Stage: 3 Challenge Level:

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

### Hockey

##### Stage: 3 Challenge Level:

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

### Pyramids

##### Stage: 3 Challenge Level:

What are the missing numbers in the pyramids?

### 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?

### 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. . . .

### More Number Pyramids

##### Stage: 3 and 4 Challenge Level:

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

### Top-heavy Pyramids

##### Stage: 3 Challenge Level:

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

### Coins on a Plate

##### Stage: 3 Challenge Level:

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

### Pattern of Islands

##### Stage: 3 Challenge Level:

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

### 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. . . .

### Concrete Wheel

##### Stage: 3 Challenge Level:

A huge wheel is rolling past your window. What do you see?

### Mediant

##### Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

### 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. . . .

### Picture Story

##### Stage: 4 Challenge Level:

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

### Leonardo's Problem

##### Stage: 4 and 5 Challenge Level:

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

### 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. . . .

### Convex Polygons

##### Stage: 3 Challenge Level:

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

### Volume of a Pyramid and a Cone

##### Stage: 3

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

### The Pillar of Chios

##### Stage: 3 Challenge Level:

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

### Eleven

##### Stage: 3 Challenge Level:

Replace each letter with a digit to make this addition correct.

### Adding All Nine

##### Stage: 3 Challenge Level:

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

### 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.

### Even So

##### Stage: 3 Challenge Level:

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

### Mindreader

##### Stage: 3 Challenge Level:

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

### No Right Angle Here

##### Stage: 4 Challenge Level:

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
user
The text passage contains a series of mathematical problems and proofs, but no explicit mathematical questions or calculations are presented in a format that requires a specific numerical answer. The content is structured as a collection of logic and reasoning problems, with each problem or proof building on mathematical concepts and challenging the reader to think critically.

TYPE: NO_QUESTION