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Here is an interactivity for Napier's Location Arithmetic.
Full Screen Version
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Part One : Pick two
numbers as multipliers (factors) to multiply together, say $37$ and
$51$. Choose some of the values from $1, 2, 4, 8, 16, 32, 64$ and
$128$ to make a sum equal to each of those numbers. For example $37
= 32 + 4 + 1$ and $51 = 32 + 16 + 2 + 1$
Incidentally, could $37$ or $51$ have been made in another way
?

Now select the side numbers needed to make the factors you've
chosen and press the "press when ready" button to begin.

Next click the counters in the grid one by one to see them move
towards the bottom line.

Finally click the counters remaining in the bottom line to
compile an answer (product) for your multiplication question.
Play with the application. Try different numbers.
Why does the process work  why does this method always
produce a correct answer?
Continue playing. Perhaps try factor numbers with particular
patterns of gaps and counters.
Part Two (quite hard)
: Are there multipliers (factors) that produce a product which has
a counter in every position along the bottom line?
Part Three (a real
challenge) : Which multiplication question requires the most
counters to represent both the multipliers (factors) and their
product (that means the number of factor counters and product
counters together making one single total).
For example $37$ uses $3$ counters at the side, $51$ uses $4$
side counters, and their product uses $9$ counters in the bottom
line, altogether a total of $16$ counters ($3 + 4 + 9$).
Sending in solutions : we would love to hear your way of
explaining why Napier's Location Arithmetic is a valid method for
multiplication, or if you make some progress with parts $2$ or $3$,
and can explain what you've done and what you've discovered, that
would be wonderful to receive also.
An addition : it's excellent
for us to hear that a problem we have offered has stimulated a new
question in someone else's mind  thankyou very much Sheldon for
this:
" I found this technique fascinating.
Is there a way of inverting the process, i.e. Factorising. Starting
with particular circles on the bottom line, and finding some
process which could create the two factors you started with " ?