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🤝 Abacus Small Friends and Big Friends: Every Formula Explained

BeadBop Guides · Updated 2026-10-14

"Friends" are the pairs of numbers that make 5 or 10. They are the rules that let a soroban add or subtract any number, even when a rod doesn't have enough free beads. Some books call big friends relatives.

Small friends (pairs that make 5)

The small friends are 1 & 4 and 2 & 3. Use them when you need to cross the heaven bead.

To add	Do this	To subtract	Do this
+4	+5 − 1	−4	−5 + 1
+3	+5 − 2	−3	−5 + 2
+2	+5 − 3	−2	−5 + 3
+1	+5 − 4	−1	−5 + 4

Example: 3 + 4. The ones rod shows 3 (three earth beads). There's only one free earth bead, so we can't push up four. Bring the heaven bead down (+5) and take away one earth bead (−1). The rod now shows 7.

Big friends (pairs that make 10)

The big friends are 1 & 9, 2 & 8, 3 & 7, 4 & 6 and 5 & 5. Use them when a rod would go past 9 or below 0.

To add	Do this	To subtract	Do this
+9	+10 − 1	−9	−10 + 1
+8	+10 − 2	−8	−10 + 2
+7	+10 − 3	−7	−10 + 3
+6	+10 − 4	−6	−10 + 4
+5	+10 − 5	−5	−10 + 5
+4…+1	+10 − 6…9	−4…−1	−10 + 6…9

Example: 7 + 6. The ones rod can't hold 13. Add 1 on the tens rod (+10), then take away 4 on the ones rod (−4). The abacus shows 13.

Combo friends (big + small together)

Sometimes the "take away" step of a big friend itself needs a small friend. 5 + 6: +10 on the tens rod, then −4 on the ones rod — but the ones rod only has the heaven bead, so −4 becomes −5 + 1. The full move is +10 − 5 + 1.

Combo	Formula
+6	+10 − 5 + 1
+7	+10 − 5 + 2
+8	+10 − 5 + 3
+9	+10 − 5 + 4
−6	−10 + 5 − 1
−7	−10 + 5 − 2
−8	−10 + 5 − 3
−9	−10 + 5 − 4

How to practise the friends

Learn one formula at a time until it is automatic, then mix them. BeadBop has a dedicated lesson for every single formula in Levels 11–14, plus a quick-fire Friend Finder game.

Quick answers

What are small friends in abacus?

Small friends are pairs of numbers that add up to 5: 1 and 4, 2 and 3. They let you add or subtract across the heaven bead.

What are big friends in abacus?

Big friends are pairs that add up to 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. They are used when a rod would go above 9 or below 0.