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.

Challenge Level

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

Challenge Level

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Challenge Level

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

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.

Challenge Level

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

Challenge Level

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

Challenge Level

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Challenge Level

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

Challenge Level

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

Challenge Level

What have Fibonacci numbers got to do with Pythagorean triples?

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?

Challenge Level

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

Challenge Level

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

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.

Challenge Level

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

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?

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?

Challenge Level

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

Challenge Level

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

Challenge Level

For how many integers ð‘› is the difference between âˆšð‘› and 9 is less than 1?

Challenge Level

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

Challenge Level

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

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?

Challenge Level

Explore the relationships between different paper sizes.

Challenge Level

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

Challenge Level

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

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?

Challenge Level

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

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.

Challenge Level

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

Challenge Level

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Challenge Level

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

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. . . .

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?