How to find a volume of this figure (which is $3080 \text{ cm}^3$) in a few seconds?

Solution 1:

I looked at the horizontal layers.

Top layer has seven, and each layer below shows seven more. So the number of cubes is $$ 7+14+\cdots+70=\frac{77}2\cdot10=385\,. $$

Solution 2:

Here's my idea for how someone could answer this in a few seconds.

1. See that there are ten horizontal layers
2. See that each layer adds seven blocks
3. Know that the tenth triangular number is $55$.
4. Know that $2 ^ 3 = 8$
5. Multiply $8 \cdot 55 \cdot 7$

Solution 3:

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.