Make 85 with four 9s

The classic four nines puzzle: build an expression from exactly four 9s that equals 85. 85 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 85 — the solver checked every possible expression.

Solutions with concatenation and decimals

Concatenation and decimals do not help for 85 either — every combination was checked.

Solutions with square roots, factorials, exponents

(sqrt(9)!! - 9) / 9 + sqrt(9)! = 85
sqrt(9)! - (9 - sqrt(9)!!) / 9 = 85
(9! / sqrt(9)!! + sqrt(9)!) / sqrt(9)! = 85
(9! / sqrt(9)! + sqrt(9)!!) / sqrt(9)!! = 85

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

How 85 compares

85 has 4 distinct solutions across all three tiers — #13 scarcest of all 100 targets. Next door, 84 has 116 solutions and 86 has 31 solutions.

If 85 went down easily, make 88 with four 9s — 2 solutions, the nearest step up in scarcity.

Make 84 with four 9s · Make 86 with four 9s

The Number Lab

Same idea, different constraints: four 9s, four 4s, or any four digits you like.

Four Nines — the daily puzzle
Four Nines solver — every target, solved
Four Fours — the 1881 original
Four Fours solver — including the impossible five
Make 24 solver — any four digits
Today's 24 — one new hand a day

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Four Nines — the daily puzzle: four 9s, one target, new at midnight
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