Weekly Problem 45 - 2015

If Sam is getting married on the 9th of November 2015 aged 30, do you know which year he was born in?

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.

There are lots of different methods to find out what the shapes are worth - how many can you find?

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Which of these games would you play to give yourself the best possible chance of winning a prize?

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?

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 ?

The Tower of Hanoi is an ancient mathematical challenge.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

If a sum invested gains 10% each year how long before it has doubled its value?

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?

How good are you at finding the formula for a number pattern ?

If you were to set the X weight to 2 what do you think the angle might be?

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.

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Weekly Problem 36 - 2007

Find the length along the shortest path passing through certain points on the cube.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

Are you at risk of being a victim of crime? How does your perception of that risk compare with the facts and figures?

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

There are two sets of numbers. The second is the result of the first after an increase by a constant percentage. How can you find that percentage if one set of numbers is in code?

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

We asked what was the most interesting fact that you can find out about the number 2009. See the solutions that were submitted.

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.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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.

Have you seen this way of doing multiplication ?

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

Can you explain what is going on in these puzzling number tricks?