How to compute the product of two positive integers:

**Step 1.** Place counters representing one factor along the bottom (horizontal) margin and counters for the other along the right-hand (vertical) margin.

**Step 2.** Place a counter in each square on the board representing the *intersection* of a column for which there is a counter in the bottom (horizontal) margin with a row for which there is a counter in the right-hand (vertical) margin. The square represents the product of those two values.

**Step 3.** Remove the counters representing the factors from both margins.

**Step 4.** Slide all counters on the board along their (SW-NE) equivalence lines *down* to the bottom horizontal row. (This is how a bishop moves in chess.)

**Step 5.** *Abbreviate* the bottom horizontal row from right to left, removing every two counters in a square and placing one counter in the square to its left. This process often causes a "chain reaction" that "carries" into squares to the left, requiring further *abbreviation*.

**Step 6.** The desired decimal product can now be obtained by converting the *abbreviated* product back to a decimal number.

Note that products represented on our \(8\times 8\) binary chessboard can be no larger than 255.

**Example:** That \(18 \cdot 13 = 234\) is illustrated below.

**Step 1.** The factor \(18=16+2\) is represented along the bottom margin and the factor \(13 = 8 + 4 + 1\) is represented along the right-hand margin.

**Step**** 2.** Counters are placed at the intersections of columns and rows for which there are counters both along the bottom margin and along the right-hand margin. For instance, a counter is placed at the intersection of the column labeled 16 at the bottom and the row labeled 4 at the right. The intersection of this column and row is a square labeled 64, and you should check that \(16\cdot 4 = 64.\)

**Steps 3 and 4.** Counters have been cleared from the bottom (horizontal) and right-hand (vertical) margins. Counters on the board have been slid down their equivalence lines to the bottom horizontal row. For instance, the only counter at a square labeled 64 has been slid along the 64-equivalence line to the 64-square in the bottom row. Likewise, the two counters that were in squares along the 16-equivalence line now both sit in the 16-square in the bottom row.

**Steps 5 and 6.** The bottom row has been abbreviated from right to left, leaving a row from which we can read \[18\cdot 13 = 128 + 64 + 32 + 8 + 2 = 234.\] The reader may wish to carry out the multiplication \(6\cdot 15,\) an example requiring extensive carries in the *abbreviation* process.

In preparation for extracting square roots on the chessboard calculator, it is worth noting here that, after obtaining \(18\cdot13=(16+2)(8+4+1)\) in Step 1 above, Step 2 amounts to an application of the distributive law to obtain:

\[(16+2)(8+4+1) = 16\cdot8 + 2\cdot8 + 16\cdot4 + 2\cdot4 + 16\cdot1 + 2\cdot1,\] with the six products on the right-hand side of this equation necessarily arranged in a 3 row by 2 column rectangle on the chessboard. More specifically, removal of one entire row and two entire columns from the second board shown above would result in a 3 row by 2 column rectangle consisting of six contiguous squares with counters in them.

Next: division on the chessboard calculator (followed by extraction of square roots)