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How many pairs of positive integers add to make one-thousand?

Using positive integers, how many different sums add to make six?

Can you discover the password to unlock the tests?

Can you find the area of the arrow?

Show how to fill a ring with the digits 1 to 8 so that no two adjacent numbers are consecutive.

How many cuboids exist for which the volume is less than 100 units

^{3}and the integer side lengths are in an arithmetic sequence?Using the digits 1, 2, 3, and 4 (no more and no less), can you make all of the integers from 1 to 25?

Can you determine the set of five numbers given the average clues?

How many different weights can you weigh using 2 kg, 3 kg and 5 kg and a set of balancing scales?

Which corner will the beam of light emerge from the prism?

For how many 2-digit numbers is the first digit greater than the second digit?

Can you find out how many birds are in the cage?

Can you work out how many people attended the party?

The most efficient method of arranging songs on a tape.

How many children are standing in the circle?

Can you would out how much ten chocolate bars would cost?

Can you discover the minimum number of children in the class with brown hair and are right handed?

Using the information given, can you work out how many of each coin I have?

How many triangles are in a fully connected pentagon?

What is the maximum amount of money that you could have in your pocket and not able to make exactly £2?

How many nines are there in all of the numbers from 1 to 100?

How many squares are there altogether on a standard chessboard?

How many triangles are there in the diagram?

How many triangles are there in the diagram?

If all the counting numbers from 1 to 1000 were written out how many digits would be written down?

Can you discover how many miles from the station did the two trains cross?

From the information given, can you work out how many schools took part in the cross country race?

What question can you ask to ensure he gets to the Cuddly-Wuddly tribe?

What is the least number of Bishop pieces required to protect every square on a 8 by 8 chessboard?

What is the sum of the visible faces on the dice?

How long will the tortoise take to catch up with the hare?

By ensuring that the row and column in the grid have the same total, how many different totals can this be done with?

How many years during the twenty-first century has a square digital sum?

Investigating products of digits.

How many 3-digit numbers exist for which the sum of the digits is six?

How many numbers below one hundred are divisible by both 2 and 3?

Which numbers that are divisible by 10 can also be written as the sum of four consecutive integers?

How many Easter eggs will be bought by the family in total?

Which lengths, using one 2 m length and two 3 m lengths can you measure directly?

Find the smallest multiple of nine containing only even digits.

What length of metal strips will be required to tile the roof of a 3x4 room?

Can you work out how old I am?

Can you discover the true weight of boy's sister?

Using four 4's (no more and no less), can you make all of the integers from 1 to 25?

What fraction of the square is shaded?

Can you work out the exact value of the fraction product?

Can you calculate how much would a passenger carrying 80 kg of luggage would be charged?

Can you find the palindrome using the clues?

How many cubes remain in the 5 × 5 × 5 cube?

Can you find the missing numerator and denominator to complete the fraction sum?

See if you can find the seed of this secret message.

How many white tiles will be needed to complete the kitchen floor design?

By adding a single unit square can you give the diagram a line of reflective symmetry?

What question can be used to logically deduce which address belongs to which parent?

Can you make 2 prime numbers using the digits 1, 2, 3 and 5?

Can you work out how far apart the trains were twenty minutes before passing each other?

How long will it take for two people to mow the lawn?

Can you discover how the number chain works?

How many 2-digit numbers have an odd product?

Can you work out who hit the centre target?

Can you work out which shape has the greatest perimeter?

On a 3x3x3 cube, how many cubes have exactly two faces painted?

Can you work out how fast Julie was travelling?

What is the largest gap between two consecutive palindromic years?

What is the minimum difference between two 5-digit pandigitals?

How many different ways can a 5x2 pathway be pathed with 2x1 paving stones?

How many peaceful queens can you place on a 4x4 chessboard?

A geometrical investigation using pentominoes.

What is the perimeter of the tenth pattern in the sequence?

Can you work out the minimum length of material required to make a frame for a photograph?

What is the biggest square based pyramid you can make from 1000 oranges?

How many ways can a £1 stamp book be filled with 9p first class and 7p second class stamps?

Investigating products that consist entirely of ones.

If you continue multiplying the digits in any 2-digit number, which starting numbers will finish on zero?

How many ways can exactly two quarter squares be shaded?

How many ways can you place a 3x2 block on a 4x3 grid?

Find the missing digit in the subtraction.

How many two digit primes can you find for which their reverse is also prime?

Explain why the hypotenuse is the longest side in a right angle triangle

How many years since the birth of Christ read the same upside-down?

Can you discover the meaning of the secret message written in the book?

What area of the squares are shaded?

How many different ways can a 2x2 grid be shaded?

By overlapping two equilateral triangles, find the area of the hexagaon

What fraction of the rectangle is shaded?

What fraction of the square is shaded?

By continuing the pattern, can you work out what fraction of a 20x20 grid will be shaded?

How many fractions with a denominator equal to 24 cannot be cancelled down?

Find the number of blocks needed to construct the 100th tower.

Can you work out when Augustus de Morgan was born?

Can you uncover the identity of the cult leader?

Without the use of any other measuring tools, how would you use a 2/3 m length of string to measure 1/2 m?

Find all 2-digit numbers that are equal to the sum of its digits product and sum

Find three different integers that add to sixteen, where the two smallest add to make the biggest.

Can you work out how much time will be saved by upgrading the computer system?

Creating diagrams using a traditional Chinese tangram.

How many matchsticks will be left if one-thousand houses are made?

Can you determine the lowest possible percentage score that the student scored in any one of the tests?

From the information, can you discover how old Sarah was when she retired?

Can you work out how many of the children are liars?

Can you split an equilateral triangle into six smaller triangles?

Can you discover the missing values in the triangular arithmagon?

Arranging triomino pieces of a 4×4 board.

Investigating the properties of the sum and product of digits.

Can you work out the combined weight of the boy, girl, and the dog?

By considering rotations and reflections to be equivalent, prove that there exists only one 3x3 magic square.

If every combination of the digits 1,2,3,4 was written down, what would be the sum of the numbers formed?

What does F

_{$n$+2}/F_{$n$}tend towards as $n$ increases?Can you work out what fraction of the diagram is shaded?

Find the width of the annulus

Find the area of the annulus.

Is the claim about the average contents of matches in a matchbox correct?

Prove that $x$ + 1/$x$ ≥ 2 for non-negative values of $x$.

What fraction of the class in total have brown eyes?

How many unique four unit cube arrangements are there?

Show how you would label two wooden cubes to display any date of the month.

Find the probability that the three white candles will be adjacent

Can you discover the dimensions of the room?

Can you work out the most efficient way to plant trees?

Using a metre stick how would you find the internal diameter of a large circular pipe?

Can you find the area of the circumscribed triangle?

How many ways can you climb ten steps?

How should the 2nd die be coloured so that there is an equal chance of getting two faces of the same colour?

Can you prove that the product of three consecutive integers, plus their mean, is always cube?

Show that two consecutive primes cannot have a sum that is double a prime.

Can you determine the 1000

^{th}term of the sequence?How far can you see from the top of the Eiffel tower?

Can you work out how many girls are at the disco?

Prove that the only number of non-overlapping squares you cannot split a unit square into are 2, 3, or 5 smaller squares.

How many edges does a dodecagon have?

How many squares can you draw on a grid measuring 4 dots by 4 dots?

If $S_n$ represents the sum of the first $n$ odd numbers, prove that $4S_n = S_{2n}$.

What is the least number which has no remainder when divided by any number from 1 to 10?

Prove that the two angles in the square are congruent.

Prove that $n$($n$ + 1)(2$n$ + 1) is divisible by six for all integer values, $n$.

Solve the equation

`a`!`b`! =`a`! +`b`!Can you work out the teacher's favourite mathematician?

How many ways can a darts player finish from 150 points?

Using the digits 1, 2, 3, 4, and 5 to form 5-digit numbers, how many are divisible by 12?

Which of the boys, 1 to 3, is most likely to guess the colour of his hat?

Can you determine the best way to construct a set of steps leading up to a platform?

Can you determine the weight of the glass in the door?

By concatenating all of the digits 1, 2, 3, 4, and 5 to form the ratio of two numbers, how many ways can you make one-half?

Find the perimeter of the hexagon.

From the clues can you work out how many lockers there are?

Can you find the height of the cone?

Find the radius of the circle inscribed inside the isosceles triangle.

Find the radius of the circle inscribed inside the right angled triangle.

Find the side length of the square inscribed inside the right angled triangle.

Can you show when the product of fractions is an integer?

Which is greater in value, the square root of two or the cube root of three?

How far above the ground do the two ladders cross?

Can you find the exact value of the letter product?

Find the probability of winning the card game

What are the dimensions of the shaded square inside the 3-4-5 right angle triangle?

What is the password to unlock all secrets?

Use the given information to find the height of the mountain.

Can you work out the missing weight?

Can you work out how old the teacher really is?

How many lines are required to construct the Mystic Rose?

By considering all the nets of a unit cube, which net has the greatest perimeter?

Prove that $x$

^{2}+ $x$ + 1 will never divide by 5.Can you work out which numbers are on the two discs?

If you had discs numbered 1 to 10, how would you separate the discs into the two bags such that no bag contains its double?

How many ways can paint a cube with two colours?

Can you crack the hacker website password?

Can you work out how many pens the girl bought and how much she paid for each one?

Can you find the height of the cone?

What is the minimum number of marks required to measure the lengths 1 to 12?

Find the sum of all 4-digit combinations taken from {1,2,3,4,5}

Finding primes that are one less than a square.

Can you prove that all primes greater than 2 can be written as the difference of two squares?

Can you prove that the square of all primes minus 1 are divisible by 24?

How many primes less than 100 can be written as the sum of two square numbers?

Prove that seven is the only prime number that is one less than a perfect cube.

Show that $a/b = c/d = (a - c)/(b - d)$.

What proportion of 3-digit numbers contain the digit one?

How would you arrange the numbers 1 to 16 in the grid, such that the product of the numbers in each quadrant is divisible by 16?

How many different routes can you find through a 4 × 4 grid?

Find the connection between the constructed length and the original rectangle.

Given the three concentric circles generated by the rectangle show that the area of the inner circle equals the ring generated by the outer circles

Find the smallest number, greater than 1, which has a remainder of 1 when divided by any of 2, 3, 4, 5, 6, or 7

Investigating the divisibility of adding a 2-digit number to its reverse.

How many times between 9 a.m. and 3 p.m. is the angle between the hour and minute hand 90

^{o}?In the given right angle triangle prove that the two marked angles are the same size

Investigate the special number machine that square roots and rounds off answers.

Finding the chance of making a rounding error.

Can you discover the connection between the input values produce the same output from two different machines?

What should the length of the straight section be on a running track be to meet IAAF requirements?

How many 3-digit numbers have two digits the same?

Explain why a number made up of the same digit can only be prime if the digit is one AND the number of digits is itself prime

Using one sand glass that measures 9 minutes and another that measures 13 minutes, how would you measure 30 minutes?

From the information given, how can the goal keeper achieve a 50% save rate?

Find the area of the lunes on the semi-circle

Find the area of the shaded cross.

Can you find the area of the shaded square inside the triangle?

For a given denominator, prove that there are always an even number of simple fractions

Find the vertical height of the slide

Find the area of the sloping square.

Can you determine the point where the pole snapped?

Can you decrypt the punchline?

Which fits better... a round plug in a square hole or a square plug in a round hole?

How many squares can you make from 240 unit tiles?

Given that [

`n`(`n`+1)(`n`+2)]^{2}= 3039162537*6, find the value of *Using rods of length 1, 2, 3, ... ,

`n`, can you construct a square?Find the smallest repunit that is divisible by 63.

Can you make the sum equal to 100?

Prove that the tangential distances PS and PT are always equal.

Prove that the sum of the given angles are 360 degrees.

Can you work out how many tiles fill the room?

Given two points, X and Y, find the area of triangle OXY

Prove that exterior angle of a triangle is equal to the sum of two opposite interior angles

How long will it take for the train to completely pass through the tunnel?

Find the minimum number of socks which must be taken from the drawers to be certain of finding five matching pairs.

Can you discover the secret of this misleading system?

Can you use a combination of logic and algebra to determine Goldilock's weight?

Can you discover what X represents?

How many solutions does the equation $|x| + 2|y| = 100$ have?

Can you prove the amazing relationship between the heights of the ladders?

Show that the alternating sign sum of squares produces triangle numbers.

Is there a secret hidden in the poetry?

Use the conjecture based on radicals to prove the Last Theorem of Fermat for $n \ge 6$.

Can you discover the radius of the ball at the bottom of the bag?

Where must the ball strike inside the square to return its point of origin?

What would be the diameter of a circular duct to feed 2, 3 or 4 circular cables?

Show that it is impossible not to form a triangle with all the edges the same colour

Prove that the segment joining the centres, $AB$, is a perpendicular bisector of the common chord $XY$.

Prove that circles determined by points on each side of a triangle and each vertex are concurrent.

Prove that there exists a sequence of $n$ consecutive composite numbers.

Prove that the product of four consecutive integers is always one less than a perfect square

What form must the constant take in the iterative formula for the limit to be integer?

Can you find the radius of the circle in the corner?

Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.

Given the three perimeters of a cuboid can you determine its volume?

Prove that 10

^{n}+1 is divisible by 11 iff`n`is oddFind the smallest number made up of the digits 1 through 9 which is divisible by 99.

Prove that there are infinitely many primitive solutions to the equation $x$

^{2}+ $y$^{2}= 2$z$^{2}.How many rectangles with integral length sides have an area equal in value to the perimeter?

With an equal chance of picking two discs the same colour, how many discs are in the bag?

Find the number of black discs in the game of chance

Prove that

`a`!`b`! =`a`! +`b`! + 2^{c}has a unique solutionCan you prove that (2

^{n})! is divisible by 2^{2n − 1}?Can you determine the depth of the well by timing how long it takes to hear the splash?

Prove that the ratio of adjacent terms in the Fibonacci sequence F

_{$n$+1}/F_{$n$}tends towards φ.Find the sum of a modified infinite Fibonacci series.

Prove that P(X = L) = P(X = L−1) for all positive integer values of L.

Prove that the sum of a proper fraction and its reciprocal can never be integer

What is the probability of winning the game of chance?

Prove that the minimum number of moves to completely reverse the positions of the coloured counters can never be square.

Prove that

`n`^{4}+ 3`n`^{2}+ 2 is never square.Find the conditions for when √

`a`−`b`= √`c`has a solution.Given a number has strictly increasing digits, what is the probability that it contains 5-digits?

Can you find the dimensions of an inscribed rectangle?

For which diagonal lengths is the area of the rectangle, which contains a unit square, integer?

Prove that log

_{$a$}($x$)log_{$b$}($y$) = log_{$b$}($x$)log_{$a$}($y$).Can you find the missing length of the trapzium?

Can you find the shaded area of the lunes?

Is the claim about the average number of matches statistically significant?

Prove that (

`a`+`b`)/2 ≥ √(`ab`)Investigate the nature of the second order recurrence relation.

Prove that the difference between the expressions $x$

^{3}+ $y$ and $x$ + $y$^{3}is a multiple of six.Determine the nature of all multiplicatively perfect numbers.

Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value.

Prove that $6^n + 8^n$ is divisible by 7 iff $n$ is odd.

Prove that in any graph there will always be an even number of odd vertices

How many differen ways can you form five primes using the digits 0 to 9?

Prove that a quadrilateral has opposite sides of equal length if and only if it is a parallelogram

Investigating the number of paving a pathway.

How many rectangles exist for which the number of tiles on the perimeter are equal to the number of tiles on the inside?

Can you determine the fraction of the regular pentagon that the star occupies?

Prove that all even perfect numbers are triangle numbers.

Prove that the constructed triangle inside the unit square is isosceles but not equilateral

Prove that there can be no more than five regular (convex) polyhedra

Prove that the roots of the polynomial, $x$

^{n}+ $c$_{n-1}$x$^{n-1}+ ... + $c$_{2}$x$^{2}+ $c$_{1}$x$ + $c$_{0}= 0, are irrational or integer.Can you prove that the expression 21

^{n}− 5^{n}+ 8^{n}is divisible by 24?Find the least value of $x$ for which x

^{x}+ 1 is divisible by 2^{n}.Prove that 8

^{n}−1 is always divisible by 7Prove that there exists no prime which is one less than a multiple of four that can be written as the sum of two squares.

Given that $a$

^{2}+ $b$ and $a$ + $b$^{2}differ by prime amount find $a$ and $b$.Given that $p$ is prime, when is $4^p + p^4$ prime?

Given that $p$ is prime, when is $2^p + p^2$ prime?

Prove that if $a^n - 1$ is prime and $n \ge 2$ then $a = 2$ and $n$ must be prime.

Prove that there only ever exists one prime value $p$ for which $p + 1$ is a perfect power of $k$ and determine the condition for this perfect power to exist.

Prove that 2

^{p}+ 3^{p}can never be a perfect square.Prove that the for every primitive Pythagorean triplet ($a$, $b$, $c$) that $abc$ is divisible by sixty.

Show that the midpoints of any quadrilateral form a parallelogram

Prove that the radical axis of two intersecting circles is their common secant.

When does the quadratic,

`ax`^{2}−`px`−`p`= 0, have rational roots?Prove the a quadratic with non-zero integral coefficients can be found for which every arrangement of coefficients yields rational roots.

Solve the Diophantine equation

`c`=`a`/`b`+`b`/`a`Prove that

`p`and`p`^{2}cannot be permutations of the value of their Totient functionProve that the sum of two 2-digit numbers, for which the reverse of their digits forms a different pair of 2-digit numbers with the same sum, is always divisible by 11.

Can you show that every term in the sequence is divisibly by 18?

Find the fraction of the square which the shaded octagon fills.

Find the radius of the shaded circle.

Find the shortest distance from one corner of a cuboid to the opposite corner.

When is 8

`p`+1 square?Prove that sum of the squares of $x$ and $y$ can never be a multiple of their product.

Prove that N is a sum-product number iff it is composite.

Prove that the sum of the first $n$ terms in the Fibonacci sequence is given by F

_{n+2}− 1.Prove that three touching circles with a common tangent hold a special relationship between their radii

Find the probability that the triangle formed by the hands on a clock is isosceles

How many trailing zeroes does 1000! contain?

Find the maximum area of the triangle region in the unit square

Prove that each median in a triangle is split in the ratio 2:1.

Find the series of the reciprocals of triangle numbers

When is 8

`p`+1 a triangle number?Given that $n$ is a positive integer and 2$n$ + 1 and 3$n$ + 1 are perfect squares, prove that $n$ is divisible by 40.

Evaluate the exact value of the series.

Given that $x$

^{2}+ $x$ + 1 = 0, find the value of $x$^{3}.Show that a unique circle must pass through the vertices of the two triangles

Are you able to complete the email challenge?

How many 5-digit ZIP codes are detour-prone?

Prove that the two smaller angles in the triangle are exactly 15

^{o}and 75^{o}respectively.Prove that cos($x$) is algebraic if $x$ is a rational multiple of Pi.

Prove that infintitely many

`almost equilateral triangles`existProve that for all composite values of $n$ > 4, F

_{n}is composite.For the given construction prove that the segments are congruent and concurrent with one another.

Prove that the product of given ratios in any triangle is always one.

Which shape is more likely to contain the origin?

Prove that the convergents generated by the recurrence relation for continued fractions are irreducible.

Prove that the recurrence relation holds for continued fractions.

How many digits does the number 2

^{1000}contain?Solve the Diophantine equation,

`x`^{2}+ (`b`−`x`)`y`= ±1Prove that Euler's rule, V + R = E + 2, is true for all planar graphs

Prove that P is an even perfect number iff it is of the form $2^{n-1}(2^n - 1)$ where $2^n - 1$ is prime.

Prove that every even $n$ ≥ 48 can be written as the sum of two abundant numbers.

Prove that the given identities will produce every primitive Pythagorean triplet.

Given that $m$ and $n$ are positive integers, solve $m$

^{n}= $n$^{m}.Solve the equation

`a`!`b`! =`a`! +`b`! +`c`^{2}Solve the factorial equation

`a`!`b`! =`a`! +`b`! +`c`!What fraction is the immediate predecessor of 2/5 in F

_{100}of the Farey sequence?Prove that the given

`n`^{th}term formula for the Fibonacci sequence is trueProve that 99

^{n}ends in 99 for odd`n`How long does the rocket take to reach its maximum height?

Given that $n$ is a positive integer, when is $n$

^{4}+ 4 prime?Show that the Gamma function is a suitable extension of the factorial function

Prove that the quotient and remainder for any prime cannot be in a geometric sequence.

Estimate the sum of the first one hundred Harmonic numbers

Show that

`px`^{2}−`qx`+`q`= 0 has no rational solutionsWhat fraction of the large red circle do the infinite set of smaller circles represent?

Integrate the integer part function of 10

^{$x$}Prove that $e$ is irrational.

Prove that π is irrational.

Prove that $\cos(1^o)$ is irrational.

Find the fixed monthly payments to repay a loan of £2000 in thirty-six months at an annual rate of 12%

Find the expected return on this game of chance.

Prove that the volume of an open-top box is maximised iff the area of the base is equal to the area of the four sides.

Investigate sets for which the sum of elements is fixed and the product is maximised.

Prove that the centres joining the constructed equilateral triangles from any triangle always form an equilateral triangle.

How many numbers below one million have increasing digits?

Prove that 14

^{n}+ 11 is never primeProve that an odd perfect number must be of the form $c$

^{2}$q$^{4$k$+1}where $q$ ≡ 1 mod 4 is prime.Prove that $p - 1$ is always a multiple of $ord(a, p)$.

Prove that the sum of pairwise products for a set of three real numbers whose sum is one cannot exceed one third

Prove that the last digit of an even perfect number will be 6 or 8.

Show that $x$

^{$n$}+ $y$^{$n$}= $p$ has no solution if $n$ contains an odd factor greater than one.Investigating the properties of a triangle for which two of its medians are perpendicular.

Prove that for every odd prime, $p$, there exists a unique positive integer, $n$, such that $n^2 + np$ is a perfect square.

Prove that there exist no set of primes for which the sum of reciprocals is integer

Prove that the given matrix transformation will transform a primitive Pythagorean triplet into a new one.

Find the co-ordinate of the point on the y-axis generated by the quadratic and the circle

Determining the values of $n$ for which the quadratic equation has no solutions.

Find the area of the shaded region generated by four overlapping quarter circles

Prove that the Pell equation $x^2 - dy^2 = 1$ can be used to find convergents for $\sqrt{d}$

Determine the probability that all

`r`random chords are non-intersecting.When is the sum of reciprocal square roots rational?

For a given

`x`, determine the number of solutions of 1/`x`+1/`y`=1/`z`If $GCD(n, 10) = 1$, prove that there exists some repunit which is divisible by $n$.

Can you find the coefficient of friction between the two surfaces?

Find the probability that three randomly chosen points of a 5 by 5 lattice will form a triangle

Prove that there exists a sum of $n$ distinct squares that is also square.

Prove that there always exists a number made up of ones and zeroes that is divisible by the positive integer,

`n`Prove that all integers greater than 28123 can be written as the sum of two abundant numbers.

Can you prove that the sum of the tangents is equal to the product?

Prove that the time the body takes to pass between two point can never be integer.

Prove that if a prime is expressible as the sum of two squares it can be done in only one way