How to find a volume of this figure (which is $3080 \text{ cm}^3$) in a few seconds?
I looked at the horizontal layers.
Top layer has seven, and each layer below shows seven more. So the number of cubes is $$ 7+14+\cdots+70=\frac{77}2\cdot10=385\,. $$
Here's my idea for how someone could answer this in a few seconds.
- See that there are ten horizontal layers
- See that each layer adds seven blocks
- Know that the tenth triangular number is $55$.
- Know that $2 ^ 3 = 8$
- Multiply $8 \cdot 55 \cdot 7$
So we start on the left, and kind of slice it diagonally, if it makes sense.
The first diagonal layer has TWO columns, one with $2$ blocks and another with $2$ blocks.
The second diagonal layer has THREE columns, one with $4$ blocks, another with $3$ blocks, and another with $3$ blocks.
The third diagonal layer has FIVE columns, with $6$, $4$, $2$, $2$ and $1$ blocks.
If we sum it up to the tenth diagonal layer, we end up with a total of $385$ blocks.
EDIT: Didn't see the pattern.