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1.Splits

Try this as an activity with kids - groups or individuals; it's suitable for a wide ability range.

Give them ten counters and ask them to put them into three piles. Inevitably some will say that it can't be done until you point out that you didn't say three equal piles...

Now ask them how many counters they put in each pile: (3,3,4), (2,3,5) and so on. The idea is to find how many different ways you can do this - clearly (2,3,5) is the same as (2,5,3) so it only counts once.

Let's call this 10 split 3. Ten counters split into three piles in however-many different ways.

So what about 10 split 4? Or 10 split 5?

Investigate.

What if you started with 9 counters? Or 11?

Hours of fun for all the family...;-)
 

Scroll down the page for the answer.


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

The results of our (limited) research is shown below. Of course, that doesn't explain why the results hold...but that's another story and your comments would be welcome.

split 1 2 3 4 5 6 7 8 9 10 11 12
1 1                      
2 1 1                    
3 1 1 1                  
4 1 2 1 1                
5 1 2 2 1 1              
6 1 3 3 2 1 1            
7 1 3 4 3 2 1 1          
8 1 4 5 5 3 2 1 1        
9 1 4 7 6 5 3 2 1 1      
10 1 5 8 9 7 5 3 2 1 1    
11 1 5 9 11 9 7 5 3 2 1 1  
12 1 6 12 15 13 10 7 5 3 2 1 1

From the table we note that the answer to the original question is 8. In other words, 10 split 3 is 8.
 

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