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#### Resources tagged with Mathematical reasoning & proof similar to Inspector Remorse:

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### There are 184 results

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

### Our Ages

##### Stage: 4 Challenge Level:

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

### Diophantine N-tuples

##### Stage: 4 Challenge Level:

Can you explain why a sequence of operations always gives you perfect squares?

### An Alphanumeric

##### Stage: 5

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

### Euler's Squares

##### Stage: 4 Challenge Level:

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

### Never Prime

##### Stage: 4 Challenge Level:

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

### 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}.

### For What?

##### Stage: 4 Challenge Level:

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

### Look Before You Leap

##### Stage: 5 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

### Dalmatians

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

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

### Polynomial Relations

##### Stage: 5 Challenge Level:

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

### Exhaustion

##### Stage: 5 Challenge Level:

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

### 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?

### Euclid's Algorithm II

##### Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

### Composite Notions

##### Stage: 4 Challenge Level:

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

### Whole Number Dynamics IV

##### Stage: 4 and 5

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

### Whole Number Dynamics I

##### Stage: 4 and 5

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

### Mod 3

##### Stage: 4 Challenge Level:

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

### Always Perfect

##### Stage: 4 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

### Find the Fake

##### Stage: 4 Challenge Level:

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

##### Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

### Basic Rhythms

##### Stage: 5 Challenge Level:

Explore a number pattern which has the same symmetries in different bases.

### Converse

##### Stage: 4 Challenge Level:

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

### 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/...)).

### 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.

### Modulus Arithmetic and a Solution to Dirisibly Yours

##### Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

### Doodles

##### Stage: 4 Challenge Level:

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### More Sums of Squares

##### Stage: 5

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

### Sums of Squares and Sums of Cubes

##### Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

### Transitivity

##### Stage: 5

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

### Fractional Calculus III

##### Stage: 5

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

### Proofs with Pictures

##### Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### 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...

### 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.

### 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.

### Mouhefanggai

##### Stage: 4

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

### Modulus Arithmetic and a Solution to Differences

##### Stage: 5

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

### A Computer Program to Find Magic Squares

##### Stage: 5

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

### The Triangle Game

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

Can you discover whether this is a fair game?

### 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.

### Knight Defeated

##### Stage: 4 Challenge Level:

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

### 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.

### 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.

### Pythagorean Triples II

##### Stage: 3 and 4

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

### Little and Large

##### Stage: 5 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

### 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!

### Proof of Pick's Theorem

##### Stage: 5 Challenge Level:

Follow the hints and prove Pick's Theorem.

### A Knight's Journey

##### Stage: 4 and 5

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

### Russian Cubes

##### Stage: 4 Challenge Level:

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

### 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. . . .