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Can you work out the equations of the trig graphs I used to make my pattern?
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?
A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?
A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
Why does this fold create an angle of sixty degrees?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you describe this route to infinity? Where will the arrows take you next?
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Which of these games would you play to give yourself the best possible chance of winning a prize?
NRICH has always had good solutions from Madras College in St Andrew's, Scotland but the solutions to this problem were truly exceptional.
It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?
Find the sum of the series.
If a sum invested gains 10% each year how long before it has doubled its value?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Use the differences to find the solution to this Sudoku.
Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
What is the same and what is different about these circle questions? What connections can you make?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...