other digit by that amount in terms of the alphabet (b + 2 => d). For example, multiply 4 = c by 16 = e c * e = 2^2 * 2^4 = 2^6 = g or... AlphabetIndex(c) = 2, so... e => f => g To find the product of two multiple digit numbers, make a two column table. In the left column write the digits of the first number, one below the other. For each digit in the left column, multiply that digit and the second number and record it in the right column. Finally, add all the numbers of the right column together. As an example, multiply 238 = bcdfgh by 13 = acd {| |- | a || bcdfgh |- | c || defhij |- | d || efgijk |} The result is the sum in the right column = = bcekl = 2+4+16+1024+2048 = 3094. It is interesting to notice that the left column can also be obtained by successive halves of the first number, from which the even numbers are removed. In our example, acd, bc (even), ab, a. Noticing that the right column contains successive doubles of the second number, shows why the peasant multiplication is exact. Division, remainder Division can be carried out by successive subtractions: the quotient is the number of time the divisor can be subtracted from the dividend, and the remainder is what is left rest after all the possible subtractions. This process, which can be very long, may be made efficient if instead of the divisor we subtract multiple of the divisor, and computations are easier if we restrict to multiple by a power of 2. In fact, this is what we do in the long division method. The grid Location arithmetic uses a square grid where each square on the grid represents a value. Two sides of the grid are marked with increasing powers of two. Any inner square can be identified by two numbers on these two sides, one being vertically below the inner square and the other to its far right. The value of the square is the product of these two numbers. For instance, the square in this example grid represents 32, as it is the product of 4 on the right column and 8 from the bottom row. The grid itself can be any size, and larger grids simply permit us to handle