Location Arithmetic - Division

Division

Martin Gardner presented a slightly easier to understand version of Napiers division method, which is what is shown here.

Division works pretty much the reverse of multiplication. Say we want to divide 485 by 13. First place counters for 485 (= 111100101) along the bottom edge and mark 13 (= 1101) along the right edge. To save space, we'll just look at a rectangular portion of the board because that's all we actually use.

Starting from the left, the game is to move counters diagonally into "columns of divisors" (that is, with one counter on each row marked with a 1 from the divisor.) Let's demonstrate this with the leftmost block of counters.

Now the next block of counters we might try would begin with the leftmost counter on the bottom, and we might attempt something like

except that we don't have any counters that we can move diagonally from the bottom edge into squares that would form the rest of the "column of divisors."

In such cases, we instead "double down" the counter on the bottom row and form a column one over to the right. As you will soon see, it will always be possible to form a column this way. So first replace the counter on the bottom with two to its right.

and then move one diagonally to the top of the column, and move another counter located on the edge of the board into its spot.

It looks like we still don't have a counter on the bottom edge to move diagonally into the remaining square, but notice that we can instead double down the leftmost counter again and then move it into the desired square.

and now move one counter diagonally to where we want it.

Let's proceed to build the next column. Once again, notice that moving the leftmost counter to the top of the column doesn't leave enough counters at the bottom to fill in the remaining squares.

So we double down the counter and move one diagonally into the next column over. Let's also move the rightmost counter into the column, and here's how it looks after these steps.

We still have a missing square, but we just double down again and move the counter into this spot and end up with

At this point, the counter on the bottom edge is so far to the right that it cannot go diagonally to the top of any column, which signals that we are done.

The result is "read" off the columns—each column with counters is treated as a 1 and empty columns are 0. So the result is 100101 (= 37) and the remainder is the binary value of any counters still left along the bottom edge. There is one counter on the third column from the right, so we read it as 100 (= 4) and we get 485 ÷ 13 = 37 with a remainder 4.