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#### Resources tagged with Mathematical reasoning & proof similar to In Between:

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### There are 186 results

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

### Plus or Minus

##### Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

##### Stage: 4

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.

### Golden Eggs

##### Stage: 5 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

### Target Six

##### Stage: 5 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

### How Many Solutions?

##### Stage: 5 Challenge Level:

Find all the solutions to the this equation.

### Pent

##### Stage: 4 and 5 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.

### Thousand Words

##### Stage: 5 Challenge Level:

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

### Continued Fractions II

##### Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

### Whole Number Dynamics III

##### Stage: 4 and 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.

### Whole Number Dynamics II

##### Stage: 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.

### Where Do We Get Our Feet Wet?

##### Stage: 5

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

### More Sums of Squares

##### Stage: 5

Tom writes about expressing numbers as the sums of three squares.

### 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.

### Magic Squares II

##### Stage: 4 and 5

An article which gives an account of some properties of magic squares.

### 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. . . .

### 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.

### Pythagorean Triples II

##### Stage: 3 and 4

This is the second article on right-angled triangles whose edge lengths are whole numbers.

### Square Mean

##### Stage: 4 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

### There's a Limit

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

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### 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. . . .

### 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.

### The Root Cause

##### Stage: 5 Challenge Level:

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

### Long Short

##### Stage: 4 Challenge Level:

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

### Three Ways

##### Stage: 5 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

### Proof of Pick's Theorem

##### Stage: 5 Challenge Level:

Follow the hints and prove Pick's Theorem.

### Pythagorean Triples I

##### Stage: 3 and 4

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

### 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.

### Square Pair Circles

##### Stage: 5 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.

### The Clue Is in the Question

##### Stage: 5 Challenge Level:

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

### Notty Logic

##### Stage: 5 Challenge Level:

Have a go at being mathematically negative, by negating these statements.

### Contrary Logic

##### Stage: 5 Challenge Level:

Can you invert the logic to prove these statements?

### Direct Logic

##### Stage: 5 Challenge Level:

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

### L-triominoes

##### Stage: 4 Challenge Level:

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

### A Long Time at the Till

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

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

### Calculating with Cosines

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

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

### Interpolating Polynomials

##### Stage: 5 Challenge Level:

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

### Dodgy Proofs

##### Stage: 5 Challenge Level:

These proofs are wrong. Can you see why?

### Advent Calendar 2011 - Secondary

##### Stage: 3, 4 and 5 Challenge Level:

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

### Iffy Logic

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

Can you rearrange the cards to make a series of correct mathematical statements?

### 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.

### 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. . . .

### Number Rules - OK

##### Stage: 4 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

### Impossible Sandwiches

##### Stage: 3, 4 and 5

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

### 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?

### Perfectly Square

##### Stage: 4 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why?

### 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?

### Symmetric Tangles

##### Stage: 4

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!