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Small friends practice: the 5-complement rules

Four 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
7
− 4
+ 6
+ 8
?
2
5
− 1
+ 2
− 3
?
3
2
+ 4
− 1
+ 4
?
4
8
+ 5
+ 4
+ 5
?
5
7
+ 2
+ 9
+ 2
?
6
5
+ 1
− 3
+ 4
?
7
8
− 1
+ 4
− 9
?
8
9
+ 2
− 9
+ 3
?
9
8
− 5
+ 9
+ 6
?
10
7
+ 2
− 9
+ 1
?

Full answer key

7 − 4 + 6 + 8 = 17
5 − 1 + 2 − 3 = 3
2 + 4 − 1 + 4 = 9
8 + 5 + 4 + 5 = 22
7 + 2 + 9 + 2 = 20
5 + 1 − 3 + 4 = 7
8 − 1 + 4 − 9 = 2
9 + 2 − 9 + 3 = 5
8 − 5 + 9 + 6 = 18
7 + 2 − 9 + 1 = 1

Practise it now

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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.

How to practise it

1Learn the four pairs cold: 1–4, 2–3, 3–2, 4–1.
2When a bead is not free, reach for the heaven bead first.
3Say the substitution out loud at first — "plus five, minus one".
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. Four 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.