Working systematically - advanced

Solve the equations to identify the clue numbers in this Sudoku problem.

The challenge is to find the values of the variables if you are to solve this Sudoku.

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

Can you work out where these 5 riders came in a not-quite-so-famous bike race?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

Five equations and five unknowns. Is there an easy way to find the unknown values?

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.

Investigate the transformations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0, -1 and +1.

How would you design the tiering of seats in a stadium so that all spectators have a good view?

A Sudoku based on clues that give the differences between adjacent cells.

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

A messenger runs from the rear to the head of a marching column and back. When he gets back, the rear is where the head was when he set off. What is the ratio of his speed to that of the column?

In 1% of cases, an HIV test gives a positive result for someone who is HIV negative. How likely is it that someone who tests positive has HIV?

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

DNA profiling is an invaluable tool for the police. However, when it comes to probability, things aren't always as straightforward as they seem.

Explore the strange geometrical properties of the Koch Snowflake.

This Sudoku problem consists of a pair of linked standard Sudoku puzzles each with some starting digits.

How high will a ball taking a million seconds to fall travel?

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

Can you rotate a curve to make a volume of 1?

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

Can you use numerical methods to solve this equation to 1 decimal place?

In this short challenge, can you use angle properties in a circle to figure out some trig identities?

An arithmetic progression is shifted and shortened, but its sum remains the same...

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

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?

Can you find a way to prove the trig identities using a diagram?

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

Two cannons are fired at one another and the cannonballs collide... what can you deduce?

At what angle should you throw something to maximise the distance it travels?

This problem explores the biology behind Rudolph's glowing red nose, and introduces the real life phenomena of bacterial quorum sensing.

After transferring balls back and forth between two bags the probability of selecting a green ball from bag 2 is 3/5. How many green balls were in bag 2 at the outset?

Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

Build series for the sine and cosine functions by adding one term at a time, alternately making the approximation too big then too small but getting ever closer.

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 2% of flights are over-booked?

Exponential functions grow pretty quickly...