Make 40 with four 9s

The classic four nines puzzle: build an expression from exactly four 9s that equals 40. 40 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 40 — the solver checked every possible expression.

Solutions with concatenation and decimals

Concatenation and decimals do not help for 40 either — every combination was checked.

Solutions with square roots, factorials, exponents

(9 + 9 × sqrt(9)) / .9 = 40
(9 - sqrt(9))! / (9 + 9) = 40
(9 / sqrt(9))!! / (9 + 9) = 40
(9 - sqrt(9)!)!! / (9 + 9) = 40
(9 / (.9 + .9))! / sqrt(9) = 40

…and 232 more at this tier.

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

How 40 compares

40 has 237 distinct solutions across all three tiers — #25 most plentiful of all 100 targets. Next door, 39 has 84 solutions and 41 has 3 solutions.

If 40 went down easily, make 39 with four 9s — 84 solutions, the nearest step up in scarcity.

Make 39 with four 9s · Make 41 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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