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Resources tagged with Mathematical reasoning & proof similar to Climbing Powers:

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

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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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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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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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More Sums of Squares

Stage: 5

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

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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.

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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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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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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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Take Three from Five

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

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?

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A Biggy

Stage: 4 Challenge Level: Challenge Level:1

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.

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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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Largest Product

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

Which set of numbers that add to 10 have the largest 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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Mod 3

Stage: 4 Challenge Level: Challenge Level:1

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

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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 orf two or more cubes.

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How Many Solutions?

Stage: 5 Challenge Level: Challenge Level:1

Find all the solutions to the this equation.

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What Numbers Can We Make Now?

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

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

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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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Advent Calendar 2011 - Secondary

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

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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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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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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Little and Large

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

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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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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A Long Time at the Till

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

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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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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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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Dodgy Proofs

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

These proofs are wrong. Can you see why?

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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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Modular Fractions

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

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

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Calculating with Cosines

Stage: 5 Challenge Level: Challenge Level:1

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

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Common Divisor

Stage: 4 Challenge Level: Challenge Level:1

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

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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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Binomial

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

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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Big, Bigger, Biggest

Stage: 5 Challenge Level: Challenge Level:1

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

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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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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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Polite Numbers

Stage: 5 Challenge Level: Challenge Level:1

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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Pair Squares

Stage: 5 Challenge Level: Challenge Level:1

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

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Prime AP

Stage: 4 Challenge Level: Challenge Level:1

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

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Telescoping Functions

Stage: 5

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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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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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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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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Mind Your Ps and Qs

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

Sort these mathematical propositions into a series of 8 correct statements.

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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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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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Iffy Logic

Stage: 4 Short Challenge Level: Challenge Level:1

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

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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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The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

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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?