Peter from Durham Johnston School wrote:

1) The rule is the difference between the last two numbers in the sequence +1 added onto the previous number = the next number in the sequence.

X = the number you are trying to find, Y and Z = the past two numbers.

Z is the previous number that is before the number you are trying to find.

A = The difference between Z and Y

The difference between Z and Y + 1 = A .

A + 1 + Z = X

To put this into a real situation:

Sequence: 4, 6 , 9 , X

9 - 6 = 3

9 + 3 + 1 = X = 13

This then continues using the same rule. Below is a graph to show the relationship between the volume and amount of pebbles.

2) The rule is the difference between the previous two numbers in the sequence + 2 + the previous number = the next number in the sequence.

X = The number you are trying to find.

Y and Z = the previous two numbers in the sequence.

The difference between Z and Y = A

A + 2 + Z = X

3) It is always going to be 0.

4) The rule is the same as Q 2.

Zach sent in a very thorough explanation. You can view it in full here: More Pebbles - Zach T.pdf and it starts off like this:

Thank you for these solutions to what was really a tough nut to crack. Well done!