Make 31 with four 9s

The classic four nines puzzle: build an expression from exactly four 9s that equals 31. 31 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 31 — the solver checked every possible expression.

Solutions with concatenation and decimals

Concatenation and decimals do not help for 31 either — every combination was checked.

Solutions with square roots, factorials, exponents

(.9 + 9 × sqrt(9)) / .9 = 31
sqrt(9)!! / (9 + 9) - 9 = 31
(99 - sqrt(9)!) / sqrt(9) = 31
sqrt(9)! × sqrt(9)! / .9 - 9 = 31
(.9 + sqrt(9) ^ sqrt(9)) / .9 = 31

…and 10 more at this tier.

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

How 31 compares

31 has 15 distinct solutions across all three tiers — #32 scarcest of all 100 targets. Next door, 30 has 455 solutions and 32 has 3 solutions.

If 31 went down easily, make 32 with four 9s — 3 solutions, the nearest step up in scarcity.

Make 30 with four 9s · Make 32 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
The Reflex Lab — nine ways to measure your hands and memory, none of them polite about it
hand me something weird — a random pick from the directory. Could be the horse.

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