Weird Multiplication

Why do the following methods work?

1. Suppose you want to calculate . Write these two numbers at the top of two columns. In the first column, start with 18, divide it by two, and repeat, ignoring any remainders, until you reach the number 1. In the second column, start with 14, double it, and repeat, until you have a column of equal length. Draw a line through any number in the first column that is even, and extend the line to cross out the neighboring number in the second column. Add up the remaining numbers in the second column. This will be the product, and this procedure works for any pair of numbers. Two examples:

   

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2. Number the fingers on each hand 5, 6, 7, 8, and 9, starting with your pinkies and ending with your thumbs. Hold your hands out in front of you with the palms facing you, thumbs up, and pinkies down.

   To multiply two numbers on the above list together, touch the corresponding fingers together. For example, to calculate , touch the index finger of your left hand to the middle finger of your right hand.

   At and above the index finger on your left hand there are 2 fingers. At and above the middle finger on your right hand there are 3 fingers. Multiply these two numbers together: . The remaining fingers on both hands combined each count 10, and there are 5 of them, for a product of 50. Add 6 and 50 to get the answer: .

   Confused? Let's try another one: . Touch the ring finger of your left hand to the index finger of your right hand. Multiplying the number of fingers at and above the touching fingers: . Counting fingers below the touching fingers as tens: 40. Add 8 and 40 to get the answer: 48.