This game is Bulls and Cows, played with words

The rule this game runs on is not new. It is not even close to new. Two numbers — how many you got in the right place, how many are present but misplaced — is the feedback rule of Bulls and Cows, a pencil-and-paper game that was old before anyone commercialised it, and that has a word version older than every daily word game on the internet.

What follows is the lineage, and then the part nobody seems to have done: measuring the three feedback rules against each other on the same word list, so “this one is harder” becomes a number instead of an opinion.

1955: Jotto, and one number

Jotto was invented in 1955 by Morton M. Rosenfeld and sold by his New York company, Jotto Corp. Two players each write down a secret five-letter word. You guess; your opponent tells you how many letters your guess has in common with their word. That is the entire reply. One number.

No positions. Not even a hint of position. Guess PEACH against OTHER and the answer is “2” — the E and the H — with nothing said about where they sit. It is a startlingly severe game, and it was popular enough that Rosenfeld advertised it on the Jack Paar and Mike Wallace shows.

Bulls and Cows, and two numbers

Bulls and Cows is the numbers cousin, of uncertain age and no clear inventor — it was a schoolyard game long before it was a product. Each player writes a secret four-digit number with no repeated digits. After each guess you are told two counts:

• Bulls — right digit, right position.
• Cows — right digit, wrong position.

Read those two definitions again with letters in place of digits and you have this game exactly. Hits are bulls. Nears are cows. Wordtallyo is Bulls and Cows played on a dictionary, with a curated answer instead of an opponent, and a tally drawn as strokes because the game is named after the tally.

We did not set out to rebuild a seventy-year-old pencil game. It is simply where the rule leads: if you want to withhold which letters while still saying something useful about position, two counts is the natural shape, and other people got there first.

1970: Mastermind makes it a product

Mordecai Meirowitz, an Israeli postmaster and telecommunications expert, invented Mastermind in 1970; Invicta Plastics released it in 1971–72 after the Nuremberg toy fair. Coloured pegs replace digits and the key pegs are bulls and cows by another name: a coloured key peg for right colour in the right hole, a white one for right colour in the wrong hole.

The purists of the day were unimpressed. A 1976 Games & Puzzles forum called Mastermind a sorry cousin to Bulls and Cows, Jotto and My Word — which tells you both how established the older games were, and that the argument about which version is the real one is itself half a century old.

Mastermind is also where the mathematics arrived. With four holes and six colours there are 6⁴ = 1,296 codes, small enough to attack exhaustively, and in 1976 Donald Knuth proved the codebreaker can always finish in five moves or fewer using a minimax rule: choose the guess whose worst case leaves the fewest possibilities. Every solver in this project, including the one that set our guess limits, is a descendant of that idea.

The three rules, measured

Here is the part that seems not to exist anywhere else. These three games differ only in what the feedback reveals, so put all three rules on the same dictionary and the same answer pool — ours, 646 five-letter answers and 8,636 legal guesses — and find the best possible opening guess under each.

A perfect opener under Jotto’s rule leaves 172.48 of 646 answers — 27% of the pool untouched, after the best guess available in the language. Bulls and Cows takes you to 12%. Coloured tiles take you to 2%.

The mechanism is the last column. Jotto’s reply can come back 5 distinguishable ways against this pool. Two counts can come back 17 ways. A colour pattern, 104. A game is only as sharp as the number of different things its feedback can say, and those three numbers are the three eras.

Where this game leaves the lineage

Three deliberate departures from Bulls and Cows, each for a reason:

• A curated answer, not an opponent’s word. The classic games are two-player, and the secret word is whatever your opponent thought of. A daily puzzle needs everyone to face the same board, so answers come from a pool filtered for words people actually know. You can still play it the old way — a seed link hands a specific word to whoever opens it.
• Repeated letters are allowed on the six-letter board. Bulls and Cows traditionally bans repeated digits, and so do our four- and five-letter boards. The hard board permits them and does not tell you when one is present, which is the single nastiest thing in the game and entirely in the spirit of the original.
• The guess limits were measured, not chosen. Knuth’s five-move proof works because 1,296 codes is small. Our pools run to 1,117 and the dictionaries to 15,232, so exhaustive proof is out and we used percentiles of a solver over every answer instead. That is how the easy board ended up with more guesses than the hard one — a result nobody expected.

Sources

Dates, inventors and rules above are from Wikipedia’s articles on Jotto, Bulls and Cows and Mastermind. The measurements are ours: npm run openers regenerates them from the word lists the game ships, and this page reads that output directly.