# 1000 Published Shorts

These problems have been adapted from the 1000 Problems to Enjoy collection.

### Point in Between

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

Find the point on the line segment AB that is twice as far from B as it is from A.

### Mean Square

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

Can you fill this square so that the number in the middle of each line is the mean of the two numbers on either side of it?

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

What is the sum of these fractions?

### Multiple Choice

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

How many questions did Sarah answer correctly in this multiple choice exam?

### Two Triangles

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

Prove that the angle marked $a$ is half the size of the angle marked $b$.

### Leg It

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

How far from the finishing line should these runners start to make the race 'fair'?

### Inner Rectangle

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

If the shape on the inside is a rectangle, what can you say about the shape on the outside?

### No Rights

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

Prove that the angle bisectors of a triangle can never meet at right angles.

### Rectangle Split

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

Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.

### Huge Powers

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

Can you put these numbers in order?

### Candy Floss

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

What length of candy floss can Rita spin from her cylinder of sugar?

### A Mean Calculation

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

What is the mean of this set of numbers?

### Similar Cylinders

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

Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?

### Loo Roll Emergency

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

When the roll of toilet paper is half as wide, what percentage of the paper is left?

### Circles on a Triangle

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

Find the missing length on this diagram.

### Pendants

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

What proportion of each of these pendants will be made of gold?

### Tunnel Vision

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

How wide is this tunnel?

### How Many Digits?

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

How many digits are there in this product?

### Folded Square

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

This square piece of paper has been folded and creased. Where does the crease meet the side AD?

### Two Equilateral Triangles

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

Prove that these two lengths are equal.

### Distance to the Corner

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

Can you find the distance from the well to the fourth corner, given the distance from the well to the first three corners?

### Half Area

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

The horizontal red line divides this equilateral triangle into two shapes of equal area. How long is the red line?

### The Property Market

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

A property developer sells two houses, and makes a 20% loss on one and a 20% profit on the other. Overall, did he make a profit or a loss?

### Snake Eyes

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

Can you find the area of the yellow part of this snake's eye?

### Domino Match

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

If you take two dominoes from a set at random, what is the probability that they 'match'?

### Ratio Cut

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

Cutting a rectangle from a corner to a point on the opposite side splits its area in the ratio 1:2. What is the ratio of a:b?

### Semicircle in a Triangle

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

A semicircle is drawn inside a right-angled triangle. Find the distance marked on the diagram.

### Diamond Ring

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

Find the radius of the stone in this ring.

### When the Boat Comes In

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

When you pull a boat in using a rope, does the boat move more quickly, more slowly, or at the same speed as you?

### Backwards Laps

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

If two friends run in opposite directions around a track, and they pass each other every 24 seconds, how long do they take to complete a lap?

### Height of the Tower

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

How do these measurements enable you to find the height of this tower?