Urdhva-Tiryagbyham: Vertically and crosswise

Urdhva-Tiryagbyham is the third sutra of Vedic mathematics. Its corollary is Adyamadyenantyamantyena.

What follows is a brief and incomplete summary of the math shortcuts this sutra contains:

Multiplying two numbers that are both two digits long
Example: multiply 12 by 34
The answer will consist of three digits: xyz
To find x, multiply the first digit in each number: 1 x 3 = 3. So the answer is three-hundred and something. This works great if you just need a quick estimation, like figuring out if you can buy a dozen items that cost 34 cents each for less than \$5.
To find y, cross multiply. Meaning you multiply the first digit of the first number by the second digit of the second number (1 x 4), and add this product (1 x 4 = 46). 4 + 6 = 10. This means that y = 0 and you have to add the 1 to x.
To find z, multiply the second digit of each number. 2 x 4 = 8
Thus, 12 x 34 = 408

Bonus shortcut: Multiplying any two-digit number by 11 in an instant.
This sutra can be used to quickly multiply, "vertically and crosswise", any two digit number by 11. Here are two examples:
36 x 11. Place a space between the two digits of the number being multiplied by eleven: 3_6. Now add the two digits: 3 + 6 = 9. Place the 9 in the space and you have the answer. 36 x 11 = 396
69 x 11. Again, place the space: 6_9. Now add the two digits: 6 + 9 = 15. The number in the center space will be 5, and you must add the 1 to the preceeding digit. Thus, 69 x 11 = 759

Multiplying two numbers that are both three digits long
This is worked in the opposite order as the two-digit method above, and may take longer than multiplying by hand.
Example: 123 x 456
Multiply the last digit of both numbers: 3 x 6 = 18. The last digit of the answer will be 8. We will carry the 1 in the next step.
Multiply crosswise the last two digits of both numbers and add the 1 carried from the previous step: (6 x 2) + (3 x 5) + 1. So 12 + 15 + 1 = 28. The second to last digit will be 8 and we will carry the 2.
Multiply the middle digit in both numbers: 2 x 5 = 10. Multiply cross wise the first and last digits: (1 x 6 = 6) and (3 x 4 = 12). Add these three products: 10 + 6 + 12 = 28. Add the 2 carried from the previous step. This yields 30. The third to the last digit will be 0 and we will carry the 3.
Multiply crosswise the first two digits of both numbers: (1 x 5) + (2 x 4) = 13. Carry the 3: 13 + 3 = 16. The fourth to the last digit will be 6 and we will carry the 1.
Last step: Multiply the first digit of both numbers: 1 x 4 = 4. Add the carried 1, making the first digit in the answer 5
Thus, 123 x 456 = 56088

Urdhva-Tiryagbyham in its converse can be used to simplify division problems involving algebraic functions. If you want to learn this method, check out the resources listed below:

RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm