What happens when you start to add up hexagonal numbers, i.e., take $1,1+7,1+7+19,1+7+19+37, \dots ?$ Which sequence do you get$?$Can you explain it using a picture of a cube$?$

Find your own patterns or relations in and among the sequences in Table $1.$ Can you explain why they happen with a picture or otherwise$?$

Can you recognise the pattern in each of the sequences in Table $3?$