# Your Solutions - Stage 4 & 5

### Network Trees

##### Age 14 to 18Challenge Level

Explore some of the different types of network, and prove a result about network trees.

### Inequalities

##### Age 16 to 18Challenge Level

Which of the statements must be true?

### Solving by Squaring

##### Age 16 to 18 ShortChallenge Level

Which of the statements is true?

### A Frosty Puddle

##### Age 16 to 18Challenge Level

Can you draw a sketch of how Frosty's volume changes as his height decreases?

### Dodgy Proofs

##### Age 16 to 18Challenge Level

These proofs are wrong. Can you see why?

### Positive Differences

##### Age 11 to 16Challenge Level

Can you fill the circles with the numbers 1 to 6?

### Maxi Pyramid

##### Age 11 to 16Challenge Level

What is the largest number you can obtain at the top of this pyramid?

### 11x11 Square

##### Age 11 to 16Challenge Level

Here's a neat trick you can do with an 11 by 11 square...

### River Crossing

##### Age 11 to 18Challenge Level

There is nothing half so much worth doing as simply messing about in boats...

### Isosceles Seven

##### Age 14 to 16Challenge Level

Is it possible to find the angles in this rather special isosceles triangle?

### IFFY Triangles

##### Age 14 to 18Challenge Level

Can you prove these triangle theorems both ways?

### The Tour De Clochemerle

##### Age 14 to 18Challenge Level

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

### Circumference Angles

##### Age 11 to 16Challenge Level

Can you prove the angle properties described by some of the circle theorems?

##### Age 11 to 16Challenge Level

Can you prove that the opposite angles of cyclic quadrilaterals add to $180^\circ$?

### Pythagoras Proofs

##### Age 11 to 16Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

### Matter of Scale

##### Age 14 to 16Challenge Level

Can you prove Pythagoras' Theorem using enlargements and scale factors?

### Overlap

##### Age 14 to 16Challenge Level

A red square and a blue square overlap. Is the area of the overlap always the same?

### Pentakite

##### Age 14 to 18Challenge Level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

### To Swim or to Run?

##### Age 16 to 18Challenge Level

The famous film star Birkhoff Maclane wants to reach her refreshing drink. Should she run around the pool or swim across?