I have problem solving the following questions:
I have infinite amount of Lego bricks with 6 and 8 studs in length. I want to build a palisade that has the length of 48 studs . In what ways can I choose Lego bricks so that it fits perfectly?
If I want to build a really long palisade, say 480 studs, in how many ways can I do that?
My thoughts: I came to the conclusion that (6 Lego bricks of 8 studs), (8 Lego bricks of 6 studs) and (3 Lego bricks of 8 studs and 4 Lego bricks of 6 studs) are the only ways to get a palisade that has the length of 48 studs. I don't know if it requires calculation to get those numbers, I just made some tests and came up with these numbers.
I have 3 cases (see above) and somehow I think that I have to see/get a pattern out of these cases and that pattern should be recursive? (I think?)
I don't know how to start solving this and I cant see any pattern from my cases. I have tried solving it as a diophantic equation but the results did not say me anything.
Any tips and advice will help!
The number of ways to partition $48$ into parts of size $6$ and $8$ is \begin{eqnarray*} [x^{48}]: \frac{1}{(1-x^6)(1-x^8)}. \end{eqnarray*} To do $480$ just replace the $48$ in the above formula.
Alternatively, the solutions of the equation $6a+8b=48$ can be parameterised by $a=4t$ and $b=3(2-t)$, so $t$ can take any value from $0$ to $2$. So there are $3$ solutions.