Problem: Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers where the first layer is one block. Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters). Solution: Image we only had to choose $4$ blocks. Then, the answer is given to us via a greedy approach. We have $1$ choice for the first block, and once we remove that , we now have $4$ choices of blocks to take. Once we pick one, we reveal $4$ new blocks giving us $7$ options of blocks to take. Taking another, we reveal $4$ more blocks and so we have $10$ blocks to choose from. Overall, we'd find that the number of ways to choose $4$ blocks is $1 \cdot 4 \cdot 7 \cdot 10$. But once we think about taking a fifth block, our approach is quickly destroyed. Insane trick: Imagine there is just one more layer, a fifth layer, and try to count all 5-block choosings. By our approach, there are $1 \cdot 4 \cdot 7 \cdot 10 \cdot 13$ such choosings. Now, we ask, how many of these block choosings had us take a block from the very last row? For this to happen, we'd have to keep 'digging' at each step choosing a newly revealed block each time. Hence, the total paths that do this are $1 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 256$. Thus, the number of $5$-block choosings that only take blocks from the first $4$ layers (which is identical to what we want to count) is $1 \cdot 4 \cdot 7 \cdot 10 \cdot 13 - 256 = 3384$.