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### Funnel

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

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 ?

### Cola Can

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

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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

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

### Equal Temperament

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

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.

### The Line and Its Strange Pair

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

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 ?

### Semi-regular Tessellations

##### Age 11 to 16 Challenge Level:

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

### A Change in Code

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

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?

### Put Out

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

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?

### Tetra Perp

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

Show that the edges AD and BC of a tetrahedron ABCD are mutually perpendicular when: AB²+CD² = AC²+BD².

### Three by One

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

There are many different methods to solve this geometrical problem - how many can you find?

### Pythagoras for a Tetrahedron

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

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . .

### Sixty-seven Squared

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

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

### Eyes Down

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

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

### Logosquares

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

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

### Telescoping Series

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

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

### Converging Product

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

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

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

Some of our more advanced investigations

### Snow and Cholera

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

What information helped medical pioneers decide on the cause of a disease? Especially in a time before microscopes were as powerful as they are today ?

### Swimmers

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

Swimmers in opposite directions cross at 20m and at 30m from each end of a swimming pool. How long is the pool ?

### Peaches in General

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

It's like 'Peaches Today, Peaches Tomorrow' but interestingly generalized.

### Unusual Long Division - Square Roots Before Calculators

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

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

### Eight Ratios

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

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

### A Scale for the Solar System

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

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

### Pythagoras’ Comma

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

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

### Six Notes All Nice Ratios

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

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.

### Mapping the Wandering Circle

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

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

### Points in Pairs

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

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

### Two Shapes & Printer Ink

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

If I print this page which shape will require the more yellow ink?

### The Rescaled Map

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

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

### Orbiting Billiard Balls

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

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

### Tin Tight

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

What's the most efficient proportion for a 1 litre tin of paint?

### Once Upon a Time

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

Can you work out the height of Baby Bear's chair and whose bed is whose if all the things the three bears have are in the same proportions?

### Tetrahedra Tester

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

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

### Xtra

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

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

### Powerful Factors

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

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

### Pythagoras Mod 5

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

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

### Factorial Fun

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

How many divisors does factorial n (n!) have?

### Cube Roots

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

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

### Halving the Triangle

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

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

### Rational Round

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

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

### Pentakite

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

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

### How Many Balls?

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

A bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.

### Chord

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

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

### The Dodecahedron

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

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

### Golden Ratio

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

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

### Shades of Fermat's Last Theorem

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

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

### Cut Cube

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

Find the shape and symmetries of the two pieces of this cut cube.

### Real(ly) Numbers

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

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?