# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof ### Top-heavy Pyramids

##### Age 11 to 14Challenge Level

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200. ### Seven Squares - Group-worthy Task

##### Age 11 to 14Challenge 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? ##### Age 11 to 14Challenge 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. . . . ### AMGM

##### Age 14 to 16Challenge Level

Can you use the diagram to prove the AM-GM inequality? ### More Number Pyramids

##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? ### Janine's Conjecture

##### Age 14 to 16Challenge 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. . . . ### One O Five

##### Age 11 to 14Challenge 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. . . . ### Aba

##### Age 11 to 14Challenge 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. ### Postage

##### Age 14 to 16Challenge 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. . . . ### Even So

##### Age 11 to 14Challenge 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? ### Chocolate Maths

##### Age 11 to 14Challenge 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. . . . ### Disappearing Square

##### Age 11 to 14Challenge 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. . . . ### Dicing with Numbers

##### Age 11 to 14Challenge 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? ### Natural Sum

##### Age 14 to 16Challenge 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. . . . ### Eleven

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge 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}. ### 1 Step 2 Step

##### Age 11 to 14Challenge Level

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps? ### Is it Magic or Is it Maths?

##### Age 11 to 14Challenge Level

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . . ### More Mathematical Mysteries

##### Age 11 to 14Challenge 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. . . . ### Elevenses

##### Age 11 to 14Challenge Level

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? ### The Triangle Game

##### Age 11 to 16Challenge Level

Can you discover whether this is a fair game? ### What Numbers Can We Make Now?

##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge 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. ### Take Three from Five

##### Age 11 to 16Challenge 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? ### Convex Polygons

##### Age 11 to 14Challenge Level

Show that among the interior angles of a convex polygon there cannot be more than three acute angles. ### Gabriel's Problem

##### Age 11 to 14Challenge Level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was? ### What Numbers Can We Make?

##### Age 11 to 14Challenge Level

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

##### Age 14 to 16Challenge Level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places? ##### Age 11 to 14Challenge 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. . . . ### Geometric Parabola

##### Age 14 to 16Challenge Level

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence. ### Sticky Numbers

##### Age 11 to 14Challenge Level

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? ### Clocked

##### Age 11 to 14Challenge 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? ##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge Level

A huge wheel is rolling past your window. What do you see? ### Yih or Luk Tsut K'i or Three Men's Morris

##### Age 11 to 18Challenge 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 Great Weights Puzzle

##### Age 14 to 16Challenge Level

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest? ### Picture Story

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 11 to 14Challenge Level

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

##### Age 11 to 14Challenge 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? ##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

I am exactly n times my daughter's age. In m years I shall be ... How old am I? ##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge 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 16Challenge 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. ### Mod 3

##### Age 14 to 16Challenge Level

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3. ### The Genie in the Jar

##### Age 11 to 14Challenge 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. . . .