Mastermind

How to crack the code

Every guess buys you two kinds of information. A filled peg means one of your shades is exactly right: correct colour, correct slot. An open peg means a shade is in the code but you have put it in the wrong place. Crucially, the feedback never tells you which of your pegs earned which mark, so the whole skill is designing guesses that let you deduce the rest. The efficient approach is to split the puzzle into two questions, which shades are in the code and where each one belongs, and to answer them in that order rather than all at once. Ten rows sounds generous, and it is: players who treat each row as an attempt to win outright burn through them and lose, while players who treat the early rows as questions rarely need more than six.

A worked opening

Watch the two-phase method run. Guess one: four pegs of the first shade. Feedback, one filled peg, so the code holds exactly one of it. Guess two: four of the second shade. Nothing comes back, and that shade leaves the puzzle entirely. Guess three: four of the third shade, two pegs, so two of those. Three rows in, your counts total three, meaning exactly one slot is still unaccounted for, and one more single-shade probe settles it. The moment the counts reach four, stop probing: any shades you never tested are ruled out for free. Now the arrangement phase begins with the game three-quarters solved and six rows still on the table.

Rearranging has its own discipline: change one thing at a time. Swap exactly two pegs between guesses and watch the filled count. Up means one of the swapped pair went home, down means you broke a match, unchanged means both or neither were placed right, and every one of those answers eliminates arrangements in bulk.

Rules of thumb that save guesses

• Spend the first guesses learning content: fill a row with a single shade, and the total number of pegs it earns tells you exactly how many of that shade the code contains.
• Once you know which shades the code uses and how many of each, stop introducing new ones and rearrange only those.
• Change one thing at a time: swap two pegs and watch whether a filled peg appears or disappears to nail down a position.
• Keep a mental list of shades already proven absent, and never waste a guess on them again.
• Track the filled-peg count across guesses; a rearrangement that raises it means you moved a shade closer to home.

The mathematics of five guesses

With six shades and four slots, repeats allowed, there are 1296 possible codes, and each guess sorts all of them into bins by the feedback it would produce. Only fourteen distinct feedback answers exist, so a well-chosen guess splits the surviving codes into small piles no matter which answer comes back, while a lazy guess leaves one giant pile intact. That is the entire theory of the game in two sentences, and it is why Donald Knuth was able to prove in 1977 that a code-breaker playing the pile-minimizing strategy wins in five guesses at worst, opening with two pairs. Later analysis trimmed the average to about 4.34 guesses. Ten tries leaves you room for a whole wasted experiment and the win besides.

The story behind Mastermind

Mastermind in its familiar peg-and-shield form was invented in 1970 by Mordecai Meirowitz, an Israeli postmaster and telecommunications expert, and the big toy companies turned it down repeatedly before Invicta Plastics, a small firm in Leicester, England, took the gamble. It became one of the defining games of the 1970s, with tens of millions of sets sold before the decade was out. The idea was not brand new even then: it is a coloured-peg descendant of Bulls and Cows, a paper-and-pencil digit game played for generations, physicalized into a hidden code behind a screen and feedback delivered in small black and white key pegs. And the famous box art, a distinguished bearded man with a woman standing behind him, starred two locals: a Leicester hairdresser named Bill Woodward and a computing student named Cecilia Fung, who posed again for anniversary editions decades later.