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#### Resources tagged with Mathematical reasoning & proof similar to Power Up:

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### There are 184 results

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

### 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.

### More Sums of Squares

##### Stage: 5

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

### Truth Tables and Electronic Circuits

##### Stage: 2, 3 and 4

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

### Logic, Truth Tables and Switching Circuits Challenge

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

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

### How Many Solutions?

##### Stage: 5 Challenge Level:

Find all the solutions to the this equation.

### 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.

### A Biggy

##### Stage: 4 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

### Largest Product

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

Which set of numbers that add to 10 have the largest product?

### Stonehenge

##### Stage: 5 Challenge Level:

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

### Three Ways

##### Stage: 5 Challenge Level:

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

### 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.

### Big, Bigger, Biggest

##### Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

### 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.

### Perfectly Square

##### Stage: 4 Challenge Level:

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

### What Numbers Can We Make Now?

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

### 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.

### 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 orf two or more cubes.

### Modular Fractions

##### Stage: 5 Challenge Level:

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

### Prime AP

##### Stage: 4 Challenge Level:

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

##### Stage: 4 Challenge Level:

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

### Without Calculus

##### Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

### 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.

### 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.

### 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...

### Russian Cubes

##### Stage: 4 Challenge Level:

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . .

### 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.

### Euler's Formula and Topology

##### Stage: 5

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

### Shape and Territory

##### Stage: 5 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

### The Triangle Game

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

Can you discover whether this is a fair game?

### Proofs with Pictures

##### Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### 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.

### Similarly So

##### Stage: 4 Challenge Level:

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

### 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?

### Circle Box

##### Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

### Generally Geometric

##### Stage: 5 Challenge Level:

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

### 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.

### Round and Round

##### Stage: 4 Challenge Level:

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

### 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.

### Rhombus in Rectangle

##### Stage: 4 Challenge Level:

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

### Matter of Scale

##### Stage: 4 Challenge Level:

Prove Pythagoras Theorem using enlargements and scale factors.

### 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.

### 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/...)).

### Telescoping Functions

##### Stage: 5

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

### 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.

### Try to Win

##### Stage: 5

Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...

### 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.

### 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 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.

### 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.