Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

Unit Interval

Age 14 to 18
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?

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?

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?

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. . . .

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.

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.

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?

N000ughty Thoughts

Age 14 to 16
Challenge Level

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

Gift of Gems

Age 14 to 16
Challenge Level

Four jewellers share their stock. Can you work out the relative values of their gems?

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. . . .

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?

Concrete Wheel

Age 11 to 14
Challenge Level

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

Leonardo's Problem

Age 14 to 18
Challenge Level

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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. . . .

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.

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. . . .

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. . . .

Eleven

Age 11 to 14
Challenge Level

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

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.

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.

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. . . .

Sixational

Age 14 to 18
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. . . .

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.

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?

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?

What Numbers Can We Make?

Age 11 to 14
Challenge Level

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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.

Iffy Logic

Age 14 to 18
Challenge Level

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

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?

Seven Squares - Group-worthy Task

Age 11 to 14
Challenge Level

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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.

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.

Largest Product

Age 11 to 14
Challenge Level

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

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?

More Number Pyramids

Age 11 to 14
Challenge Level

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

Children at Large

Age 11 to 14
Challenge Level

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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?

Number Rules - OK

Age 14 to 16
Challenge Level

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

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

The Triangle Game

Age 11 to 16
Challenge Level

Can you discover whether this is a fair game?

Magic Squares II

Age 14 to 18

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

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. . . .

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. . . .

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.

Chocolate Maths

Age 11 to 14
Challenge Level

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Dicing with Numbers

Age 11 to 14
Challenge Level

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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.

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.