In how many different ways can 24 identical boxes (cuboids), each measuring 1 x 1 x 2 be packed into a single (cuboid) container measuring 2 x 2 x 12? Mirroring and rotation do not count.
This is a math problem from school (I´m in grade 9). I tried to define possible sub-arrangements of, say four of the 1 x 1 x 2 cuboids. These sub-arrangements are either "closed" or have elements protruding to the next "layer". Yet somehow this doesn´t get me any further. I appreciate your advice.
We are building $(2\times2\times n)$-towers with these cuboids. A full tower of height $n$ is legally filled up to height $n$, and has $r$ vertical cuboids sticking out one unit on top. Make sure that the only possible values for $r$ are $0$, $2$, $4$, whereby in case $r=2$ the two vertical cuboids share a side wall. Denote by $a_r(n)$ $\>(r\in\{0,2,4\})$ the number of full towers of height $n$ having $r$ vertical cuboids sticking out on top.
It is not totally clear what you mean by "rotations and mirroring do not count". In my counting we now have $$a_0(1)=2,\quad a_2(1)=4,\quad a_4(1)=1\ .$$ The main work is to set up a recursion scheme $$a_0(n+1)=\ldots,\quad a_2(n+1)=\ldots,\quad a_4(n+1)=\ldots\quad.$$ Once you have that there is the "Master Theorem" that allows you to find closed expressions for the $a_r(n)$.