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### Cannon Balls

##### Age 16 to 18 Short Challenge Level:

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

### Root Hunter

##### Age 16 to 18 Challenge Level:

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

### Negatively Triangular

##### Age 14 to 16 Challenge Level:

How many intersections do you expect from four straight lines ? Which three lines enclose a triangle with negative co-ordinates for every point ?

### Overarch 1

##### Age 16 to 18 Short Challenge Level:

This short question asks if you can work out the most precarious way to balance four tiles.

### Log Attack

##### Age 16 to 18 Challenge Level:

Solve these equations.

### T for Tan

##### Age 16 to 18 Challenge Level:

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

### Spotting the Loophole

##### Age 14 to 16 Challenge Level:

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?

### Sine and Cosine

##### Age 14 to 16 Challenge Level:

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?

### Take a Message Soldier

##### Age 14 to 18 Challenge Level:

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?

### Belt

##### Age 16 to 18 Challenge Level:

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

### Terminology

##### Age 14 to 16 Challenge Level:

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

### Unit Interval

##### Age 14 to 18 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

##### Age 14 to 16 Challenge Level:

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. . . .

### Triangle Midpoints

##### Age 14 to 16 Challenge Level:

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

### Lunar Leaper

##### Age 16 to 18 Challenge Level:

Gravity on the Moon is about 1/6th that on the Earth. A pole-vaulter 2 metres tall can clear a 5 metres pole on the Earth. How high a pole could he clear on the Moon?

### From Point to Point

##### Age 14 to 16 Short Challenge Level:

Can you combine vectors to get from one point to another?

### Googol

##### Age 16 to 18 Short Challenge Level:

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

### Sixinit

##### Age 16 to 18 Short Challenge Level:

Choose any whole number n, cube it, add 11n, and divide by 6. What do you notice?

### Moving Stonehenge

##### Age 16 to 18 Challenge Level:

A look at the fluid mechanics questions that are raised by the Stonehenge 'bluestones'.

### Stereoisomers

##### Age 16 to 18 Challenge Level:

Put your visualisation skills to the test by seeing which of these molecules can be rotated onto each other.

### Uniform Units

##### Age 14 to 16 Challenge Level:

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

### The Genes of Gilgamesh

##### Age 14 to 16 Challenge Level:

Can you work out the parentage of the ancient hero Gilgamesh?

### Using the Haemocytometer

##### Age 14 to 16 Challenge Level:

Practise your skills of proportional reasoning with this interactive haemocytometer.

### Bio Graphs

##### Age 14 to 16 Challenge Level:

What biological growth processes can you fit to these graphs?

### Coin Lines

##### Age 14 to 16 Challenge Level:

If a coin rolls and lands on a set of concentric circles what is the chance that the coin touches a line ?

### Counting Fish

##### Age 14 to 16 Challenge Level:

I need a figure for the fish population in a lake. How does it help to catch and mark 40 fish?

### Jumping Gerbils

##### Age 14 to 16 Challenge Level:

A conveyor belt, with tins placed at regular intervals, is moving at a steady rate towards a labelling machine. A gerbil starts from the beginning of the belt and jumps from tin to tin.

### Zeller's Birthday

##### Age 14 to 16 Challenge Level:

What day of the week were you born on? Do you know? Here's a way to find out.

### Universal Time, Mass, Length

##### Age 16 to 18 Short Challenge Level:

Can you work out the natural time scale for the universe?

### Equal Equilateral Triangles

##### Age 14 to 16 Challenge Level:

Using the interactivity, can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

### The Fire-fighter's Car Keys

##### Age 14 to 16 Challenge Level:

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

### Under the Ribbon

##### Age 14 to 16 Challenge Level:

A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?

### ' Tis Whole

##### Age 14 to 18 Challenge Level:

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

### Wedge on Wedge

##### Age 14 to 16 Challenge Level:

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

### The Mean Problem

##### Age 11 to 14 Short Challenge Level:

Can you work out the four unknown numbers from the clues about their means?

### Couples

##### Age 11 to 14 Challenge Level:

In a certain community two thirds of the adult men are married to three quarters of the adult women. How many adults would there be in the smallest community of this type?

### Nutty Mixture

##### Age 7 to 11 Challenge Level:

Use the ratio of cashew nuts to peanuts to find out how many peanuts Rachel has. What would the ratio be if Rachel and Marianne mixed their bags?

### Biggest Enclosure

##### Age 14 to 16 Challenge Level:

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

### Obviously?

##### Age 14 to 18 Challenge Level:

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

### Dirisibly Yours

##### Age 16 to 18 Challenge Level:

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

### 396

##### Age 14 to 16 Challenge Level:

The four digits 5, 6, 7 and 8 are put at random in the spaces of the number : 3 _ 1 _ 4 _ 0 _ 9 2 Calculate the probability that the answer will be a multiple of 396.

### Watching the Wheels Go 'round and 'round

##### Age 7 to 11 Challenge Level:

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

### Home City

##### Age 7 to 11 Challenge Level:

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

### Two Circles

##### Age 14 to 16 Challenge Level:

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

### Dividing the Field

##### Age 14 to 16 Challenge Level:

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

### Burning Down

##### Age 14 to 16 Challenge Level:

One night two candles were lit. Can you work out how long each candle was originally?

### Gambling at Monte Carlo

##### Age 14 to 16 Challenge Level:

A man went to Monte Carlo to try and make his fortune. Is his strategy a winning one?

### There and Back

##### Age 14 to 16 Challenge Level:

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

### Seriesly

##### Age 16 to 18 Challenge Level:

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

### Upsetting Pitagoras

##### Age 14 to 18 Challenge Level:

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2