Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

Number Rules - OK

Age 14 to 16
Challenge Level

Can you produce convincing arguments that a selection of statements about numbers are true?

Power Mad!

Age 11 to 14
Challenge Level

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

Tis Unique

Age 11 to 14
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.

Janine's Conjecture

Age 14 to 16
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. . . .

Adding All Nine

Age 11 to 14
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. . . .

Aba

Age 11 to 14
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.

Perfectly Square

Age 14 to 16
Challenge Level

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

One O Five

Age 11 to 14
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. . . .

Even So

Age 11 to 14
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?

Triangle Incircle Iteration

Age 14 to 16
Challenge Level

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

Archimedes and Numerical Roots

Age 14 to 16
Challenge Level

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Mindreader

Age 11 to 14
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. . . .

Multiplication Square

Age 14 to 16
Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Mod 3

Age 14 to 16
Challenge Level

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

Common Divisor

Age 14 to 16
Challenge Level

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

A Biggy

Age 14 to 16
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.

Postage

Age 14 to 16
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. . . .

Why 24?

Age 14 to 16
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.

Eleven

Age 11 to 14
Challenge Level

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

What Numbers Can We Make Now?

Age 11 to 14
Challenge Level

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

Euler's Squares

Age 14 to 16
Challenge Level

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

Gabriel's Problem

Age 11 to 14
Challenge Level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Konigsberg Plus

Age 11 to 14
Challenge Level

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Tourism

Age 11 to 14
Challenge Level

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Disappearing Square

Age 11 to 14
Challenge Level

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

Convex Polygons

Age 11 to 14
Challenge Level

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

Proof Sorter - Quadratic Equation

Age 14 to 18
Challenge Level

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

To Prove or Not to Prove

Age 14 to 18

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

Dalmatians

Age 14 to 18
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.

Top-heavy Pyramids

Age 11 to 14
Challenge Level

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

Picture Story

Age 14 to 16
Challenge Level

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

Go Forth and Generalise

Age 11 to 14

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.

Clocked

Age 11 to 14
Challenge Level

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

Concrete Wheel

Age 11 to 14
Challenge Level

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

Tessellating Hexagons

Age 11 to 14
Challenge Level

Which hexagons tessellate?

More Mathematical Mysteries

Age 11 to 14
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. . . .

Natural Sum

Age 14 to 16
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. . . .

More Number Pyramids

Age 11 to 14
Challenge Level

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

N000ughty Thoughts

Age 14 to 16
Challenge Level

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

Our Ages

Age 14 to 16
Challenge Level

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

DOTS Division

Age 14 to 16
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}.

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

Composite Notions

Age 14 to 16
Challenge Level

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

Yih or Luk Tsut K'i or Three Men's Morris

Age 11 to 18
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. . . .

Königsberg

Age 11 to 14
Challenge Level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Magic Squares II

Age 14 to 18

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

Mediant Madness

Age 14 to 16
Challenge Level

Kyle and his teacher disagree about his test score - who is right?

The Triangle Game

Age 11 to 16
Challenge Level

Can you discover whether this is a fair game?

The Genie in the Jar

Age 11 to 14
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. . . .

Tower of Hanoi

Age 11 to 14
Challenge Level

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.