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Resources tagged with Mathematical reasoning & proof similar to Route to Root:

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

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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.

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Ordered Sums

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Russian Cubes

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Dalmatians

Stage: 4 and 5 Challenge Level: Challenge Level:1

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.

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Power Mad!

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Postage

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Knight Defeated

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Triangle Incircle Iteration

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . .

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Archimedes and Numerical Roots

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Magic W Wrap Up

Stage: 5 Challenge Level: Challenge Level:1

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.

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Recent Developments on S.P. Numbers

Stage: 5

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

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Plus or Minus

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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Doodles

Stage: 4 Challenge Level: Challenge Level:1

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not. . . .

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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...

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N000ughty Thoughts

Stage: 4 Challenge Level: Challenge Level:1

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

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Three Frogs

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Unit Interval

Stage: 4 and 5 Challenge Level: Challenge Level:1

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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Perfectly Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Water Pistols

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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Some Circuits in Graph or Network Theory

Stage: 4 and 5

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

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Pareq Exists

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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Proof: A Brief Historical Survey

Stage: 4 and 5

If you think that mathematical proof is really clearcut and universal then you should read this article.

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Cube Net

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Golden Eggs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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Janine's Conjecture

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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. . . .

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The Root Cause

Stage: 5 Challenge Level: Challenge Level:1

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

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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.

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Magic Squares II

Stage: 4 and 5

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

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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.

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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.

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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?

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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.

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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.

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Picturing Pythagorean Triples

Stage: 4 and 5

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

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Whole Number Dynamics I

Stage: 4 and 5

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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Can it Be

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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Sixational

Stage: 4 and 5 Challenge Level: Challenge Level:1

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. . . .

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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.

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Number Rules - OK

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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...

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Euclid's Algorithm II

Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

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Quadratic Harmony

Stage: 5 Challenge Level: Challenge Level:1

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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Proof of Pick's Theorem

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Follow the hints and prove Pick's Theorem.

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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.

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Square Pair Circles

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Thousand Words

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Here the diagram says it all. Can you find the diagram?

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The Clue Is in the Question

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

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Diverging

Stage: 5 Challenge Level: Challenge Level:1

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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Converse

Stage: 4 Challenge Level: Challenge Level:1

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .

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Notty Logic

Stage: 5 Challenge Level: Challenge Level:1

Have a go at being mathematically negative, by negating these statements.