Make 31 with four 4s

The classic four fours puzzle: build an expression from exactly four 4s 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 4s 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

(4 + 4!) / 4 + 4! = 31
((4! / 4)! + 4!) / 4! = 31
(.4 + 4! / sqrt(4)) / .4 = 31
(4! - sqrt(4)) / .4 - 4! = 31
((sqrt(4) / .4)! + 4) / 4 = 31

…and 4 more at this tier.

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

How 31 compares

31 has 9 distinct solutions across all three tiers — #21 scarcest of the 95 solvable targets. Next door, 30 has 453 solutions and 32 has 599 solutions. The nearest impossible target is 73, 42 away — proven unreachable, not just unsolved.

If 31 went down easily, make 33 with four 4s — 4 solutions, the nearest step up in scarcity.

Make 30 with four 4s · Make 32 with four 4s

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

Got what you came for?

Stay for something you did not search for.

Four Nines — the daily sibling: 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.

Read up on the 1881 original, pick another number in the solver, or find more free math games.