Square Roots on Napier’s Checkerboard

I’ve figured out how to do square-roots in binary on Napier’s Checkerboard!

I’ve learned how to do addition, subtraction, multiplication and division from James Tanton’s vids. I’ve shown how to do addition, subtraction and multiplication before, so now I’ll show division:


Let’s do 250 / 13 on Napier’s Checkerboard.

Division is just reverse-engineering multiplication, so we’ll treat this as an answer to a multiplication problem. Now we need to work out the picture from it.

So, what’s the structure of the board in a multiplication problem? It’s one where some of the rows are matching, and some of the columns are also matching. The rest of the rows/columns are empty. The ones that match represent the factors of the multiplication problem.

So our goal is to get the 8, 4 and 1 rows matching, and the rest empty. How can we do that?

The answer is 19 (bottom row) and 3 is the remainder (black checkers).

Wait…what? We’re still on division? Here’s how to do square-roots.


Let’s do 121 on Napier’s Checkerboard. This is similar to division, because we need to reverse engineer squaring. So, ok, thats’ cool: all we need to do is divide where we don’t know what the divisor is! 😉

Now, the problem is: how do we even start? We don’t know what to divide by! But I do know one thing:

The “resulting picture” must be symmetric about the diagonal.

Like, ok, look at 11²:

It’s symmetric along the diagonal!

The only other thing we know is that the picture must have the “multiplication structure” I outlined earlier. This is what makes it “square-looking”. The corner of the square represents the biggest value, so…wait, we can do something now!

Slide the leftmost counter (64) up until it hits the diagonal.

Ok, so that counter is now done, which I’ll mark in the next picture. Now we can do the same with 32, right?

Well, no. If we try to slide the 32 counter up to the diagonal, it will just “skip over” it. So, ok, can we just let it skip over? No, then it’s not symmetrical. So, here’s what I’ll do:

I’ll turn that one counter into two counters one place to the right…
…and then slide each of them up: one just under the diagonal, one just over the diagonal.

Ok, 2 more counters are now done, but now look what happened. The 8 and the 2 rows aren’t the same anymore! We can fix this by adding another counter to the (2,2) spot:

Ok, you might not understand what I’ve done here at first glance, but look: I’ve turned the 8 counter into two 4 counters, and slid one up.

Ok, so that’s one more counter done. Now we can just repeat what we have done! Take the 16 counter, and turn it into two 8 counters. Then slide them up so that out rules are not broken:

If you are wondering why I slid one up 3 positions, and not one 1 position and the other 2 positions, it’s because that will break our “multiplication structure” rule. Don’t get me wrong, this also does, but we can fix it: we do the same as before! We take the 4 counter, and split it up into two 2 counters. Then we slide one up…

…et voilà! All of the counters are now done!

Now we can just read straight off that the square root of 121 (which was the number we began with) is 11!

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