Bubbles inside Bubbles inside Bubbles inside ...

 

Bubbles can be separate or inside other bubbles. They do not touch or join together (or burst!).
This diagram shows the 4 different ways that three bubbles can be blown:
Here are some of the ways that four bubbles can be blown:
1 Find other ways of blowing four bubbles. Have you found all possible ways?
 
2 Copy and complete this table of results :
Number of bubbles (N)
1
2
3
4
...
Number of ways (B)

 

3 Extend your results table to five bubbles. Try to be systematic when drawing your bubble pictures. Finding all the possibilities is not easy.
(Hint: It may be easier trying to add a fifth bubble to each four-bubble picture. Where are all the possible places the fifth bubble could go? Work through each picture in sequence. Watch out for repeats though!)
 
4 This diagram shows some Code Numbers:
1
Can you work out the code numbers for all your five-bubble pictures?
 
5 Not all numbers are Code Numbers. They obey rules. Can you explain why the following numbers are not code numbers for five-bubble pictures:
11113 21111 12311 22322
Try to make up the rules that Code Numbers should obey.
 
6 Make a list of the 9 Code Numbers for all the four-bubble pictures. Write the list in a column in increasing order.
 
 
Try to work out a way of getting each code number from the previous one.
Now make a list of your five-bubble Code Numbers in increasing order. Work through your list carefully trying to find any missing Code Numbers. If you find any then draw the corresponding Picture.
 
7 How many ways of blowing six bubbles are there? Try to use Code Numbers to find out.
 
8 Is there a formula that tells you the number of ways ,W, that you can blow B bubbles?
 
Investigate.
Go to Amazing Number Facts No 14 for inspiration