larger numbers. Notice that moving either one square to the left or one square up doubles the value. This property can be used to perform binary addition using just a single row of the grid. Addition First, lay out a binary number on a row using counters to represent the 1s in the number. For example, 29 (= 11101 in binary) would be placed on the board like this: The number 29 is clearly the sum of the values of the squares on which there are counters. Now overlay the second number on this row. Say we place 9 (= 1001 in binary) on it like this. The sum of these two numbers is just the total value represented by the counters on the board, but some of the squares have more than one counter. Recall however, that moving to the left of a square doubles its value. So we replace two counters on a square with one counter to its left without changing the total value on the board. Note that this is the same idea used to abbreviate location numerals. Let's start by replacing the rightmost pair of counters with a counter to its left, giving: We still have another square with two counters on it, so we do it again: But replacing this pair created another square with two counters on it, so we replace a third time: Now each square has just one counter, and reading off the result in binary 100110 (= 38) gives the correct result. Subtraction Subtracting is not much more complicated than addition: instead of adding counters on the board we remove them. To "borrow" a value, we replace a counter on a square with two to its right. Let's see how we might subtract 12 from 38. First place 38 (= 100110 in binary) on a row, and then place 12 (= 1100 in binary) under it: For every counter on the lower row that has a counter above it, remove both counters. We can remove one such pair on the board, resulting in: Now we need to "borrow" counters to get rid of the remaining counter on the bottom. First replace the leftmost counter on the top row with two to its right: Now replace one of the two counters with two more to its right, giving: We can now take away one of the counters on the top row