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Four Fours

The four fours puzzle: make every number from 1 to 100 with exactly four 4s. Rules, the 1881 history, and the five targets that provably cannot be done.

The rules

  • Use exactly four 4s in every expression — no more, no fewer, no other digits.
  • Start with addition, subtraction, multiplication, division, and parentheses.
  • When a target refuses to fall, escalate: concatenation (44), decimals (.4), then square roots and factorials.
  • The classic challenge is every number from 1 to 100. Five of them cannot be done. Knowing which five is half the fun.

Example solutions

Every target from 1 to 100 has its own page with verified solutions — open the Four Fours solver to pick one, or go straight to 73, the most famous impossible one.

In print since 1881

The four fours puzzle asks one question: using exactly four 4s and mathematical operations, write an expression for a given whole number. 4 ÷ 4 + 4 − 4 is 1. 4 × 4 + 4 + 4 is 24. Then you try 10, discover that plain arithmetic refuses to produce it, and the puzzle has you.

It is the oldest puzzle of its kind with a paper trail. The December 30, 1881 issue of Knowledge — a popular-science magazine edited by the astronomer Richard A. Proctor — printed the problem of expressing numbers with four 4s. By 1892, when W. W. Rouse Ball included it in Mathematical Recreations and Essays, he could already file it under 'traditional': people had evidently been passing it around long before anyone thought to write it down.

The standard challenge is to make every number from 1 to 100. With + - × ÷ and parentheses alone you cannot get far, so the traditional rules also allow sticking 4s together (44, 444), decimals (.4), square roots, and factorials. Under those rules almost everything falls — but not everything. Exactly five targets between 1 and 100 hold out: 73, 77, 87, 93, and 99. They are reachable only with heavier machinery, such as the repeating decimal .4… (which equals 4/9) or the gamma function. And once you allow logarithms, a known formula produces any number at all from four 4s — which is exactly why serious versions of the puzzle ban logarithms.

The solver behind these pages enumerated every legal expression with exact fraction arithmetic — no floating point, no 'close enough'. When a page here says 73 cannot be done, that is a verified fact about a finite search space, not folklore.

FAQ

What is the four fours puzzle?

A classic math puzzle: using exactly four 4s and mathematical operations, write an expression that equals a given number. For example, (4 + 4) / (4 + 4) = 1 and 4 × 4 + 4 + 4 = 24. It first appeared in print in Knowledge magazine in 1881 and was already called a traditional recreation by 1892.

Which numbers cannot be made with four 4s?

Between 1 and 100, exactly five: 73, 77, 87, 93, and 99. That is with the traditional toolkit — + - × ÷, parentheses, concatenation like 44, decimals like .4, square roots, factorials, and exponents. Reaching those five requires devices most rule sets ban, such as the repeating decimal .4… (equal to 4/9) or the gamma function. Our solver proves the arithmetic, concatenation, and decimal cases exhaustively, and its advanced-operator search found nothing either — consistent with a century of failed attempts.

What operations are allowed in four fours?

Depends how strict your table is. The basic game is + - × ÷ with parentheses. The traditional extensions add concatenation (44, 444), decimals (.4, 4.4), square roots, and factorials. Our solver reports solutions in exactly those tiers, so you can see which targets need which tools — and logarithms are banned, because a known log trick makes every number trivially reachable.

How is Four Fours different from Four Nines?

Same recreation, different digit, different personality: four 4s is the 1881 original with five famously impossible targets, while four 9s can make every number from 1 to 100. Four Nines also runs as a daily puzzle game on this site; Four Fours is presented here as the classic it is, with a full verified solver.

Is there a four fours solver?

Yes — the solver on this site has verified solutions for every target from 1 to 100, organized by which operations they need, plus proofs of impossibility for the five holdouts. Each number has its own page where you can try it yourself before revealing the answers. Free in your browser, no download, no sign up.

The Number Lab

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

Or browse more free math games.