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Resources tagged with Mathematical reasoning & proof similar to More Secret Transmissions:

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

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.

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?

Basic Rhythms

Stage: 5 Challenge Level:

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

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.

Magic W Wrap Up

Stage: 5 Challenge Level:

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

Postage

Stage: 4 Challenge Level:

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

N000ughty Thoughts

Stage: 4 Challenge Level:

How many noughts are at the end of these giant numbers?

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. . . .

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.

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.

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?

Sixational

Stage: 4 and 5 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Prime AP

Stage: 5 Challenge Level:

What can you say about the common difference of an AP where every term is prime?

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.

Big, Bigger, Biggest

Stage: 5 Challenge Level:

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

Ordered Sums

Stage: 4 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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.

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. . . .

Triangle Incircle Iteration

Stage: 4 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

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.

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.

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}.

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.

Logic, Truth Tables and Switching Circuits

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 record your findings in truth tables.

Binary Sequences

Stage: 5 Challenge Level:

Show that the infinite set of finite (or terminating) binary sequences can be written as an ordered list whereas the infinite set of all infinite binary sequences cannot.

Symmetric Tangles

Stage: 4

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

Cube Net

Stage: 5 Challenge Level:

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

Perfectly Square

Stage: 4 Challenge Level:

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

Why 24?

Stage: 4 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Largest Product

Stage: 3 and 4 Challenge Level:

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

More Sums of Squares

Stage: 5

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

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...

Take Three from Five

Stage: 3 and 4 Challenge Level:

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Stage: 5 Challenge Level:

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

Truth Tables and Electronic Circuits

Stage: 3, 4 and 5

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

Tree Graphs

Stage: 5 Challenge Level:

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

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.

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

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.

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.

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.

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...

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.

Pythagorean Golden Means

Stage: 5 Challenge Level:

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

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.

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. . . .

Matter of Scale

Stage: 4 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

Round and Round

Stage: 4 Challenge Level:

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

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.