Four, and you are asked for three more. All four earth beads are already up against the bar. Adding 3 means raising three more of them, and there are none left. The rod is not full — it reads 4, not 9 — but the beads you want are gone. This exact moment is what the small friend exists for. 4 + 3 = ?
Interactive walkthrough · Elementary A
Small friends on the soroban: move the beads yourself
The 5-complements, one rule at a time. Read what the rule is, watch it worked forwards and backwards, then take the abacus and land the rod on the number being asked for. Nine chapters, twenty-two steps, nothing timed.
What this page asks of you
This is not a drill sheet and not an article. Every chapter is a frame of a real abacus with an explanation beside it, and the last third hands the beads over: the panel names a number, you move the beads, and it tells you whether the rod reads it. A miss costs nothing — the beads stay where you left them, and after two tries the finished equation appears.
Small friends is the whole subject of Elementary A, the third rung of the nine-level ladder. Nothing is needed before it except being able to set and read a single digit on one rod.
What is the small friends (5-complement) rule?
A rod carries one heaven bead worth five and four earth beads worth one. When you need to add and there are not enough free earth beads to do it, add five instead and take back the difference. When you need to subtract and there are not enough raised earth beads, take five and give back the difference. The difference is always the partner that makes five.
| Number on the rod | Its partner |
|---|---|
| 1 | 4 |
| 2 | 3 |
| 3 | 2 |
| 4 | 1 |
There are only two pairs to know, 1 with 4 and 2 with 3, which is why this table is the complete rule rather than a sample of it. Five itself has no small friend: it is already one bead.
Move the beads yourself
The frame is showing 4.
The whole lesson, written out
Act I — why the rule exists
Five is one bead. Drop the heaven bead to the bar and the rod reads 5 with nothing below the bar moved. That single bead is the whole trick: it lets you add or remove five in one motion, whatever the earth beads happen to be doing.
Four and one. Four earth beads up is one short of five. So 1 is the small friend of 4, and 4 is the small friend of 1 — adding 4 is "add five, take one back", and adding 1 with no room is "add five, take four back". 4 + 1 = 5
Three and two. The other pair, and the last one. Three earth beads up is two short of five, so 3 partners 2. Two pairs is the complete table; there is nothing else here to memorise. 3 + 2 = 5
Act II — worked both ways
Start at four. Four earth beads at the bar, heaven bead up. The ask is +3 and the direct move is impossible, so reach for the partner: 3 and 2 make 5. 4 + 3 = 7
Add the five. Bring the heaven bead down. Nothing below the bar moves, so the rod jumps from 4 to 9 in one motion. You have added five where you wanted three — two too many. 4 + 5 = 9
Give back the two. Drop two earth beads away from the bar and the rod settles on 7. Adding five and taking two is adding three, and it took two possible moves instead of one impossible one. 9 − 2 = 7
Start at three. Three earth beads up, one still free. The ask is +4 and only one bead is free, so the same rule applies with the other pair: 4 and 1 make 5. 3 + 4 = 7
Add the five. Heaven bead down and the rod reads 8. One too many this time, not two — which is exactly why a second example is worked. The rule is not "always two", it is "always the partner". 3 + 5 = 8
Give back the one. One earth bead away from the bar and the rod reads 7. Two different pairs, the same two moves, and both problems land on the same answer. 8 − 1 = 7
Seven, and you are asked to take three. Seven is the heaven bead plus two earth beads. Removing 3 means dropping three raised earth beads, and only two are raised. The same wall, approached from the other side. 7 − 3 = ?
Read the rule backwards. If adding three was "add five, take two back", then taking three is "take five, add two back". Same pair, opposite direction. This is the half most learners are never shown, and it is why subtraction feels harder than it is. 7 − 3 = 4
Take the five. Send the heaven bead back up. The rod drops from 7 to 2 in one motion. You have removed five where you wanted three — two too many gone. 7 − 5 = 2
Give back the two. Raise two earth beads and the rod reads 4. Taking five and adding two is taking three. 2 + 2 = 4
Start at six. Six is the heaven bead plus one earth bead. Taking 4 needs four raised earth beads and there is one. The partner of 4 is 1, so this is "take five, add one back". 6 − 4 = 2
Take the five. Heaven bead up and the rod reads 1. One too few now — you removed five where you wanted four. 6 − 5 = 1
Give back the one. Raise one earth bead and the rod reads 2, in two moves that were both possible. 1 + 1 = 2
Act III — your turn
Add the five. The rod starts at 4 and the beads are live. Do only the first half of the move: bring the heaven bead down so the rod reads 9. From 4 there is exactly one way to get there, which is why this step on its own proves you used the small friend rather than counting. 4 + 5 = 9
Now give back the two. The rod is at 9. Drop two earth beads so it reads 7. That finishes 4 + 3 — done with a five and a correction, not by finding three beads that were never there. 9 − 2 = 7, so 4 + 3 = 7
Take the five. The rod starts at 7. Send the heaven bead back up so it reads 2. The same move as before, running the other way. 7 − 5 = 2
Now give back the two. The rod is at 2. Raise two earth beads so it reads 4, and the subtraction is done. If the rod lands somewhere unexpected on the way, leave it and carry on — nothing here is scored, and "Put the beads back" resets the step. 2 + 2 = 4, so 7 − 3 = 4
Three rods, no target. Every taught frame above used a single rod, because small friends is a one-rod technique. Here are three, with nothing to hit. Set a number, take some away, try a pair you have not used yet. The reading underneath follows whatever you do.
Where to go from here
Same technique, four different jobs. This page was the walkthrough; these are the others, and none of them replaces it.
The other walkthroughs
Five techniques, one ladder. Each is its own walkthrough with its own beads to move, and they are listed here in the order a learner meets them.