Triangle numbers

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.

Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?

Sam sets up displays of cat food in his shop in triangular stacks. If Felix buys some, then how can Sam arrange the remaining cans in triangular stacks?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?

This activity creates an opportunity to explore all kinds of number-related patterns.

Can you find any two-digit numbers that satisfy all of these statements?