Multiplying on a counting board is similar to Egyptian Multiplication. You do it by
doubling, multiplying by 10, and adding.* For example, to
multiply LXXXIII (83) by XXVI (26) on a counting board, you could do it
like this. You can get the XX (20) times it by doubling X (10) times it
or doubling it and then multiplying that by 10. For the VI (6) times it
you will need IIII (4) times it and II (2) times it. So here's how you
could do it.
First represent LXXXIII (83) on the counting board like this. (See How to Add and Subtract with a Counting Board.)
Then double it by replicating each of the counters on their line or
space.
Neaten this up by regrouping. There are more than 4 counters on the I
line, so we need to take 4 of them off and promote one of them to the
space above. There are more than 4 counters in the X lines, so we need
to take 4 of them of and promote one to the space above the X line.
This gives us three counters in the space above the X line, so one
needs to be taken off and another needs to be promoted up to the C
line. Now the counter board should look like this.
Since this is one of the numbers we are going to need to add, I'll put
it away safely on the other side of the counting board. Then since we
need XX (20) times LXXXIII (83) we'll get that from this by
transferring each counter up a line on the board and then the board
will look like this.
The left side here represents our stored away II (2) times LXXXIII (83)
and the right side represents XX (20) times LXXXIII (83). Now put away
the counters on the right for safe keeping on the left side and get
IIII (4) times LXXXIII (83) by doubling the first bunch of counters we
have on the left side and the board will look like this.
Again, we need to neaten this up by regrouping things. There are two
counters in the space above the I line, so one needs to be removed and
the other moved up to the X line. There are two counters on the space
above the X line, so one needs to be removed and the other needs to be
moved up to the C line. After doing this the board should look like
this.
Now these are the numbers that need to be added to get our answer, so
we can push them all together onto the right side and see what
regrouping needs to be done.
Now after we regroup this should give us our answer. There are five
counters on the X line, so we remove four and move one up to the space
above. Then with three counters on the space above the X line one needs
to be removed and another needs to be moved up to the C line. This then
makes six counters on the C line, so four need to be removed and one
moved up to the space about the C line. Then two counters will be on
the space about the C line, so one needs to be removed and the other
needs to be moved up to the M line. So now the board should look like
this.
Representing the number MMCLVIII (2158) as the answer.
*They also often multiplied by 5 by multiplying by 10 and then
halving.
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