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Location arithmetic
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===Some properties of the grid=== Unlike addition and subtraction, the entire grid is used to multiply, divide, and extract square roots. The grid has some useful properties utilized in these operations. First, all the squares on any diagonal going from the bottom left to the top right have the same value. {| border="0" cellpadding="0" cellspacing="1" style="text-align:center; background-color: white; color: black;" |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|256 | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 32 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"|256 | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|16 | style="background-color: white;" | 16 |- style="height:30px; background-color: silver;" | style="width:30px;"|256 | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|16 | style="width:30px;"| | style="background-color: white;" | 8 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|16 | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 4 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|16 | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 2 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"|16 | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 1 |- style="height:30px; background-color:white; color: black;" | style="width:30px;"|32 | style="width:30px;"|16 | style="width:30px;"|8 | style="width:30px;"|4 | style="width:30px;"|2 | style="width:30px;"|1 |} Since a diagonal move can be broken down into a move to the right (which halves the value) followed by a move up (which doubles the value), the value of the square stays the same. In conjunction with that diagonal property, there is a quick way to divide the numbers on the bottom and right edges of the grid. {| border="0" cellpadding="0" cellspacing="1" style="text-align:center; background-color: white; color: black;" |+ '''32 Γ· 8''' |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic diagonal half.svg]] | style="background-color: white;" | '''32''' |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic diagonal.svg]] | style="width:30px;"| | style="background-color: white;" | 16 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic diagonal.svg]] | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 8 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;background-color: gray;"| | style="width:30px;"|β | style="width:30px;"|β | style="width:30px;"|β | style="background-color: white;" | '''4''' |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic vertical.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 2 |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic vertical half.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| | style="background-color: white;" | 1 |- style="height:30px; background-color:white; color: black;" | style="width:30px;"|32 | style="width:30px;"|16 | style="width:30px;"|'''8''' | style="width:30px;"|4 | style="width:30px;"|2 | style="width:30px;"|1 |} To perform 32Γ·8, one locates the dividend 32 along the right side and the divisor 8 on the bottom edge of the grid. Extending a diagonal from the dividend to the square where it intersects a vertical line of the divisor, the quotient lies at the right end of the grid from this square, in this example 4. This works as moving along the diagonal does not change the value; the value of the square on the intersection is still the dividend. Thus, the dividend is the product of the squares along the bottom and right edge. Since the square on the bottom edge is the divisor, the square on the right edge is the quotient.
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