Make 84 with four 9s

The classic four nines puzzle: build an expression from exactly four 9s that equals 84. 84 cannot be made with + - × ÷, concatenation, or decimals (the solver checked every expression) — solutions need square roots, factorials, or exponents.

Solutions using only + - × ÷

No expression made of four plain 9s and + - × ÷ reaches 84 — the solver checked every possible expression.

Solutions with concatenation and decimals

Concatenation and decimals do not help for 84 either — every combination was checked.

Solutions with square roots, factorials, exponents

9 × 9 + 9 / sqrt(9) = 84
99 - 9 - sqrt(9)! = 84
99 - sqrt(9)! - 9 = 84
9 × 9 + 9 - sqrt(9)! = 84
9 + 9 × 9 - sqrt(9)! = 84

…and 111 more at this tier.

116 distinct solutions found in total, after removing trivial rearrangements.

Make 83 with four 9s · Make 85 with four 9s

The Number Lab

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