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Cosmic Abacus
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Small friends practice: the 5-complement rules

Two pairs to learn — 1 and 4, 2 and 3 — and suddenly every single-digit sum works, even when the bead you want is already up.

  • 4 rows × 1 digits, add & subtract
  • About 8 min
  • Free, no account

Sample problems

Real problems from this drill, with answers. Nothing to install, no account, no email — work them on paper, on a physical abacus, or in your head.

  1. 1
    7
    − 4
    + 6
    + 8
    ?
  2. 2
    5
    − 1
    + 2
    − 3
    ?
  3. 3
    2
    + 4
    − 1
    + 4
    ?
  4. 4
    8
    + 5
    + 4
    + 5
    ?
  5. 5
    7
    + 2
    + 9
    + 2
    ?
  6. 6
    5
    + 1
    − 3
    + 4
    ?
  7. 7
    8
    − 1
    + 4
    − 9
    ?
  8. 8
    9
    + 2
    − 9
    + 3
    ?
  9. 9
    8
    − 5
    + 9
    + 6
    ?
  10. 10
    7
    + 2
    − 9
    + 1
    ?
Full answer key
  1. 7 − 4 + 6 + 8 = 17
  2. 5 − 1 + 2 − 3 = 3
  3. 2 + 4 − 1 + 4 = 9
  4. 8 + 5 + 4 + 5 = 22
  5. 7 + 2 + 9 + 2 = 20
  6. 5 + 1 − 3 + 4 = 7
  7. 8 − 1 + 4 − 9 = 2
  8. 9 + 2 − 9 + 3 = 5
  9. 8 − 5 + 9 + 6 = 18
  10. 7 + 2 − 9 + 1 = 1

This page is one drill. The app is the whole programme.

Kani Math is a full mental-arithmetic course, not a single exercise. Free to start, no signup, and it runs on the web, Android and iPhone.

  • ✓Nine levels, from the first bead to division
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About this drill

A small friend is a pair of numbers that adds to five. To add 4 when there are not four earth beads free, you add the heaven bead (five) and take away one — because 4 is 5 minus 1. The hand does two movements, the mind does none. This substitution is the first genuinely abstract idea in the method, and it is the reason abacus training transfers to mental arithmetic at all.

The drills below stack four single digits with both signs, which guarantees several complement moves per problem. Work them slowly at first: the goal is for the substitution to become a reflex of the fingers, not a calculation the child performs consciously each time.

It is worth being exact about what makes that abstract, because from the outside it looks like a trick. Until this point every instruction was carried out literally — asked for two, the child moved two beads. Now the instruction is add three, the hand does something else entirely (five down, two back), and the answer is right anyway. To make that move at all, the child has to hold 3 as five minus two rather than as three things. That is the moment a number stops being a quantity you count and becomes a value you can take apart and rebuild, and every later technique, up to arithmetic on an imagined frame, is the same habit at a larger size.

Parents ask what progress looks like here, and at this level it is unusually easy to see. For a week or two the child pauses at every blocked digit, mouths the substitution, and the two movements come out as two separate decisions. Then in some session the pause is simply gone: the beads move in one continuous motion, the words stop, and the hesitation relocates to reading the next row instead. If your child is not there yet, the rule is worked in both directions at kani-math.com/learn/small-friends, which ends by handing the frame over — four graded steps where the learner moves the beads to hit a target, then a free three-rod board. A child who is not yet fluent at setting and reading numbers should start at kani-math.com/abacus/how-to-use; this page assumes the instrument and teaches only the rule.

How to practise it

  1. 1Learn the four moves cold: 1–4, 2–3, 3–2, 4–1.
  2. 2When a bead is not free, reach for the heaven bead first.
  3. 3Say the substitution out loud at first — "plus five, minus one".
  4. 4Drop the words once the fingers move without them.
Where this sits

Level 3 — Elementary A

Small friends — the 5-complement rules — on 1–2 digit problems.

Teaches: 5-complement (small friends) technique, 1–2 digit operations

Find your level with the free placement exam →

Questions

What exactly is a "small friend"?
A pair summing to five: 1 and 4, 2 and 3. They are called friends because each one substitutes for the other whenever the direct bead move is blocked.
Do small friends work for subtraction too?
Yes, mirrored. To subtract 4 when you cannot, subtract five and add one. Same pairs, opposite direction.
Should my child memorise or derive them?
Memorise. Deriving costs time on every move, and the whole point is to remove thinking from the hand. Two pairs is a very small memory load.
How is this different from big friends?
Small friends borrow from the five-bead on the same rod. Big friends borrow ten from the rod to the left. Different rod, different rule.
My child adds with small friends but freezes on subtraction.
Very common, and it is not a memory failure — the rule runs backwards and so do the fingers. In the addition form both beads travel downwards: the five-bead comes down to the bar to add five, and an earth bead drops away from it to give one back. In the subtraction form both travel upwards: the five-bead goes up to remove five, and earth beads rise to the bar to give part of it back. Drill 7 minus 3 and 6 minus 4 on their own for a few sessions — take five, give back two, and take five, give back one. Both are worked bead by bead at kani-math.com/learn/small-friends.
Will this help with the arithmetic they do at school?
The part that transfers is not speed, it is the decomposition. A child who adds four as five minus one has stopped treating numbers as counts and started treating them as things that come apart and go back together, which is what school curricula call number bonds to five — the beads simply get there first, and physically. What does not transfer on its own is written layout: a soroban child may work 7 minus 3 instantly and still need showing where the answer goes on a worksheet.
Learn the techniqueSmall friends (5-complements)Walk through it on the beads →