Finger rules

Every move is made with one of two fingers, in a fixed order. Abax shows the finger with every move and follows the order below.

Which finger

The standard Japanese method uses two fingers of one hand:

The thumb only pushes up. Everything else is the index finger.

Kojima’s six ordering rules

When one digit needs both a heaven-bead move and an earth-bead move, or needs a second rod, the order is fixed. Kojima (1954, chapter III) gives six rules; Japanese school materials teach the same.

1. Adding 6 to 9 directly (for example 1 + 6): bring the heaven bead down and push the earth beads up at the same time, as one pinch.
2. Subtracting 6 to 9 directly (for example 9 − 7): earth beads down first, then the heaven bead up.
3. Five-complement addition (for example 4 + 3, done as +5 −2): heaven bead down first, then the earth beads down, in quick succession.
4. Five-complement subtraction (for example 7 − 3, done as +2 −5): earth beads up first, then the heaven bead up.
5. Ten-complement addition (for example 4 + 8, done as −2 then +1 on the left): finish the unit rod first, then carry 1 onto the tens rod.
6. Ten-complement subtraction (for example 11 − 7, done as −1 on the left then +3): borrow 1 from the tens rod first, then work the unit rod.

Across rods the rule is simpler: numbers are entered from the highest place to the lowest, left to right, one rod at a time, carrying and borrowing as you go. A carry always lands on a rod you have already visited, never on one still to come.

Some English tutorials narrate the carry first (“add 10, take away 3”). The hand order taught in Japan, and used by Abax, is the one above: complement first and carry second when adding, borrow first and complement second when subtracting.

The move notation

The manual writes moves on one rod like this:

So “4 + 3” on a rod showing 4 is +5, −2; “4 + 8” is −2, +1 left.