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Counting Counters

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2



Take a counter and surround it by a ring of other counters that MUST touch two others. How many counters do you need to do this?
 
Imagine surrounding this ring with more counters.
How many more counters are needed now?
How many counters will there be altogether?

Were you right?

What about a bigger ring? And another? And another? 

How many counters will there be in the $9$th ring?
How many do you think there will be in the $100$th ring?
How would you predict how many counters there will be in any ring?