The Capel's garden

 Page 1


Capel Family

Capel Family (detail)

Arthur Capel, 1st Baron Capel and his Family
by Cornelius Johnson, oil on canvas, circa 1640

CANVAS & STRETCHER

The Capel's Garden

Look at the portrait of the Capel family. Behind them you can see their garden.

Here is a plan of the paths in the garden.


Capel's Garden

All these lines are paths

Can one of the Capel children walk along every path without ever having to go on the same path twice?

  yes / no (tick one)

Can they walk along every path without going on the same path twice in one quarter of their garden, like this?

Capel's Garden

  yes / no (tick one)

 

 

 

 

 

 


Page 2

Here are lots of drawings of parts of their garden, giving some of the paths. For some of these you can walk on all the paths without going on any twice (except to cross over a path). This is an easy one

For some of these drawings you would have to go on some paths twice.

start here Capel's Garden

Try out all the drawings and tick "yes" if you can get round them without going twice on a path. Tick "no" if you can't get round without going twice on a path. The arrow shows you where you should enter the drawing.  

 

a
Capel's Garden
yes / no

b
Capel's Garden
yes / no

  c
Capel's Garden
yes / no

d
Capel's Garden
yes / no

 

 

 

 

 


Page 3

e
Capel's Garden
yes / no

  f
Capel's Garden
yes / no

g
Capel's Garden
yes / no

h
Capel's Garden
yes / no

i
Capel's Garden
yes / no

j
Capel's Garden
yes / no

 

 

 

Page 4

Extension

There is an easy rule which tells you whether a pattern of paths (called a network) can be walked along without going on any path twice (this is called traversing a network).

You are going to discover this rule yourself. You need to think about how many paths meet at different points in the network.

For example, here

Capel's Garden

4 paths meet,
so we call it a 4 meeting point

At

Capel's Garden

3 paths meet;
you could go 3 different ways from it

A corner like

Capel's Garden

is the meeting of 2 paths

Go back through the ten networks you have just done, and at every meeting point write in the number of paths that meet, like this:

Capel's Garden  

 

 

Each time you have 3 or 5 paths meeting, it is an odd meeting point. Where 2 or 4 paths meet is an even meeting point. For finding your rule, the number of odd meeting points in each network is important.

 

 

 

 

 

Page 5

Extension 2

Using the information from your 10 networks now fill in the following chart; the first one has been done for you.


network letter Can you go round the network without going twice on a path? (write in yes or no) How many odd meeting points does the network have?
  a yes 2
  b    
  c    
  d    
  e    
  f    
  g    
  h    
  i    
  j    

Look at the number of odd meeting points - what makes the "yes" networks different from the "no" networks?

 

 

Complete the rule:
A network can be traversed if the number of odd meeting points is

  or none at all



Now you have found the rule, can you explain why the Capels can't go all the way round the paths in their garden without going on the same path twice?

 

 

 

 

 

 

 

Page 6


Some networks can be traversed so that you end up at the same point that you started at. Which of these two networks is like that?

Capel's Garden

Capel's Garden

By designing some more networks can you find the rule for networks where you get back to your starting point again?


The rule is

 

There is also a simple rule about where you should enter a network that can be traversed - do you have to start at an odd meeting point or at an even meeting point? Look at the networks you have already done and make this sentence correct:

To traverse a network you need to enter it at an odd / even meeting point.

Now design some more gardens for the Capel family where they can go along every path without going on the same path twice. See how complicated can you make your designs.