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Broad Topics > Numbers and the Number System > Powers & roots

Take a Square

Stage: 4 Challenge Level:

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

Function Pyramids

Stage: 5 Challenge Level:

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

Power Countdown

Stage: 4 Challenge Level:

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

Em'power'ed

Stage: 5 Challenge Level:

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

Stage: 3 and 4 Challenge Level:

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

Smith and Jones

Stage: 4 Challenge Level:

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

Weekly Challenge 16: Archimedes Numerical Roots

Stage: 5 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?

The Root of the Problem

Stage: 4 and 5 Challenge Level:

Find the sum of the series.

Lost in Space

Stage: 4 Challenge Level:

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?

Equal Temperament

Stage: 4 Challenge Level:

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

Route to Root

Stage: 5 Challenge Level:

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this. . . .

Rachel's Problem

Stage: 4 Challenge Level:

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

Deep Roots

Stage: 4 Challenge Level:

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

Ab Surd Ity

Stage: 5 Challenge Level:

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

Square Pair Circles

Stage: 5 Challenge Level:

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.

Weekly Challenge 34: Googol

Stage: 5 Short Challenge Level:

Find the smallest value for which a particular sequence is greater than a googol.

Fibonacci Fashion

Stage: 5 Challenge Level:

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

Plus or Minus

Stage: 5 Challenge Level:

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

Fit for Photocopying

Stage: 4 Challenge Level:

Photocopiers can reduce from A3 to A4 without distorting the image. Explore the relationships between different paper sizes that make this possible.

Napier's Location Arithmetic

Stage: 4 Challenge Level:

Have you seen this way of doing multiplication ?

Unusual Long Division - Square Roots Before Calculators

Stage: 4 Challenge Level:

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

Perfectly Square

Stage: 4 Challenge Level:

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

Pythagorean Fibs

Stage: 5 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

Consecutive Squares

Stage: 4 Challenge Level:

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Negative Power

Stage: 4 Challenge Level:

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

Rational Roots

Stage: 5 Challenge Level:

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Number Rules - OK

Stage: 4 Challenge Level:

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...

Archimedes and Numerical Roots

Stage: 4 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?

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

Staircase

Stage: 5 Challenge Level:

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

The Root Cause

Stage: 5 Challenge Level:

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.)

Rationals Between...

Stage: 4 Challenge Level:

What fractions can you find between the square roots of 65 and 67?

Enriching Experience

Stage: 4 Challenge Level:

Find the five distinct digits N, R, I, C and H in the following nomogram

Giants

Stage: 5 Challenge Level:

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

In Between

Stage: 5 Challenge Level:

Can you find the solution to this algebraic inequality?

Mod 7

Stage: 5 Challenge Level:

Find the remainder when 3^{2001} is divided by 7.

Root to Poly

Stage: 4 Challenge Level:

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Guesswork

Stage: 4 Challenge Level:

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.