# Three-digit multiplication puzzle, part II: ever heard of senary?

Followup to: Three-digit multiplication puzzle

I'd never heard of "senary" either, until creating this series of three puzzles. It seems that "senary" is the word for base-six notation. Who knew? Without further ado, here's part II:

Place different three-digit senary numbers (000-555) on each of the seven nodes of the following diagram: There are six lines in the diagram, on each of which are three of the nodes. On each of those lines, the result of multiplying the numbers on those nodes should end in 001 in senary. The same thing for the circle, on which are also three of the nodes.

One possible arrangement for the solution not given by the OP is:

and we can note:

$$025_6+531_6=1000_6$$
$$101_6+455_6=1000_6$$
$$231_6+325_6=1000_6$$

leaving $$155_6$$ unpaired.

• Correct! Actually, there is at least one more answer (that isn't just a rearrangement of this or of the other one). – deep thought Dec 5 '18 at 19:12

[OP is self-answering with the 'easy' answer, but there is at least one other. I will accept the first correct one.] The answer to the previous decimal question uses the numbers 250±1, 500±1, 750±1, and 999, as shown on the left. And this was basically the only answer: all possible answers to the previous question are rearrangements of the same seven numbers. So, you might guess the answer for the senary case is 130±1, 300±1, 430±1, and 555, as shown on the right...

...and yes, that is an answer. But unlike the previous question, in the senary case, it is not unique. There is at least one answer which is not just a rearrangement of this, that is to say, it uses at least one number different from the above seven.