The Capel's garden
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Arthur Capel, 1st Baron Capel and his Family 
CANVAS & STRETCHER The 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? 
yes / no (tick one) 
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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 
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 
b 
c 
d 
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e 
f 
g 
h 
i 
j 
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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 

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

3 paths meet; you could go 3 different ways from it 
A corner like 

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: 

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. 
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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: 
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? 
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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? 

By designing some more networks can you find the rule for networks where you get back to your starting point again?
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. 
Join and Support
Canvas & Stretcher:
Lining up time
William Jones & circles
William Hogarth's self portrait