Multiplication by crossing lines

Text

A visual way to multiply: for $213\times 122$ you draw groups of parallel lines for the digits of the first factor ($2,\,1,\,3$) and groups of crossing lines for the digits of the second ($1,\,2,\,2$); counting the intersections along diagonals (and carrying any tens) gives the digits of the product. Apply the method and compute $213\times 122$.

Solution

Each intersection between the $10^{p}$ group of the first factor and the $10^{q}$ group of the second contributes with weight $10^{p+q}$. Collecting by weight:

$$\begin{aligned} 10^4:\ &2\cdot 1 = 2\\ 10^3:\ &2\cdot 2 + 1\cdot 1 = 5\\ 10^2:\ &2\cdot 2 + 1\cdot 2 + 3\cdot 1 = 9\\ 10^1:\ &1\cdot 2 + 3\cdot 2 = 8\\ 10^0:\ &3\cdot 2 = 6 \end{aligned}$$

No column exceeds $9$, so no carrying is needed and we read directly $213\times 122 = \boxed{25986}.$