Matchsticks
Reasoning about the number of matches needed to build squares that
share their sides.
Imagine a matchstick.  Image

How many more are needed to make a square? How many more need adding to make yet another square along side it? Carry on adding more squares ... 

How many matches have you added? How many matches are there when you have made 10 squares in the row? 20 squares? 50 squares? 
This solution comes from Henry. Well done!
To make a square, we need three more matchsticks.
To make a second square, we need three more matchsticks.
I noticed that for each square we make we need three matchsticks, in addition to the one we had at the beginning. Therefore to make 10 squares we need a total of 31 matchsticks. For 20 squares we need 61 matchsticks and for 50 squares we need 151.
The general pattern isif $n$ is the number of squares, we use $3n + 1$ matchsticks in total.