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Location arithmetic
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===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: {| border="0" cellpadding="0" cellspacing="1" style="text-align:center; background-color: white; color: black;" |- style="height:30px; background-color: silver;" | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="background-color: white; color: black;"|'''38''' |- style="height:30px; background-color:white; color: black;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="width:30px;"| | '''12''' |} 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: {| border="0" cellpadding="0" cellspacing="1" style="text-align:center; background-color: white; color: black;" |- style="height:30px; background-color: silver;" | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|β | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| |- style="height:30px; background-color:white; color: black;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"|β | style="width:30px;"| | style="width:30px;"| |} 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: {| border="0" cellpadding="0" cellspacing="1" style="text-align:center; background-color: white; color: black;" |- style="height:30px; background-color: silver;" | style="width:30px;"|β | style="width:30px;"|[[Image:Location arithmetic two counters.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| |- style="height:30px; background-color:white; color: black;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| |} Now replace one of the two counters with two more to its right, giving: {| 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;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"|[[Image:Location arithmetic two counters.svg]] | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| |- style="height:30px; background-color:white; color: black;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| |} We can now take away one of the counters on the top row with the remaining counter on the bottom row: {| border="0" cellpadding="0" cellspacing="1" style="text-align:center; background-color: white; color: black;" |+ '''11010 = 26''' |- style="height:30px; background-color: silver;" | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| | style="width:30px;"|[[Image:Location arithmetic one counter.svg]] | style="width:30px;"| |- style="height:30px; background-color:white; color: black;" | style="width:30px;"| | style="width:30px;"| | style="width:30px;"|β | style="width:30px;"| | style="width:30px;"| | style="width:30px;"| |} and read off 26, the final result.
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