Problem Solving · Free

Deductive Reasoning

Crack the hidden colour code

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The in-app how-to demo for the Deductive Reasoning

What is the Deductive Reasoning?

Code-breaking is one of the classic games of pure deduction. In the board game invented by Mordecai Meirowitz in 1970, one player hides a row of coloured pegs, and the other tries to work it out by guessing. After each guess, small markers say how many pegs are the right colour in the right spot and how many are the right colour in the wrong spot, but not which ones. Every guess is a small experiment, and cracking the code means reasoning from all the results together. Psychologists have used the game to study deductive reasoning and hypothesis testing, and mathematicians to study the best possible strategies: Donald Knuth showed in 1977 that the classic four-peg, six-colour code can always be cracked in five guesses.

Deductive Reasoning is the Neuroptimize take on the classic code-breaking game. Each session is a few hidden codes, and you crack each one within a limited number of guesses, scoring more for guesses to spare. Beginners get a marker under every peg, which makes the clues easy to read. As you level up the game moves to the classic count-only markers, then colours that repeat, empty holes, a seventh colour and finally five-peg codes. Every peg has its own shape as well as its colour, so the game is easy to read with colour blindness.

How to play the Deductive Reasoning

The covered row at the top hides the code. Below it, the guesses you have made build up one row at a time, and the empty slots at the bottom are your next guess.
Tap the colour keys to fill the slots from left to right. Tap a filled slot, or Undo, to take a peg back. When the row is full, tap Check. On a keyboard, the number keys place colours, 0 places an empty hole, Backspace takes a peg back and Enter checks.
Each checked guess gets markers. A filled disc means a peg is the right colour in the right spot; a hollow ring means a right colour in the wrong spot. A colour in your guess only scores as many markers as the code actually holds.
At levels 1 to 7 the markers sit under each peg, and a small cross marks a colour that is not in the code at all, so you can see exactly which peg earned which marker. From level 8 they become the classic markers of the board game: a count beside the row, in no particular order.
Crack the code within the guesses allowed to score it, with bonus points for every guess left over. If you run out of guesses, the code is revealed and counts as a miss. Cracked codes in a row build a combo.
There is no time limit, but finishing quickly earns a small speed bonus. A session is 6 codes at levels 1 to 3, 4 codes at levels 4 to 7, 3 codes at levels 8 and 9 and 2 bigger codes from level 10, so it takes about the same few minutes at every level.
Difficulty rises tier by tier: 3 pegs and 4 or 5 colours at first, 4 pegs from level 4, 6 colours from level 6, classic markers from level 8, repeated colours from level 10, empty holes from level 13, a seventh colour from level 19, and 5 pegs (with 4 colours and empty holes) from level 25. The number of guesses allowed drops from 10 to 9 at levels 16 to 18 and again from level 28.

Strategies and tips

Make every guess fit the clues

Before checking, test your guess against every earlier row: if it were the code, would each earlier guess have scored exactly the markers it got? A guess that fits all the clues can win outright, and one that does not wastes a row. The results screen shows what share of your guesses fitted every clue.

Change a little at a time

When the markers are hard to read, change just one or two pegs from a previous guess. Any change in the markers then points to the pegs you moved, which makes the clue easy to interpret.

Open with a spread of colours

An opening guess that uses several different colours tells you quickly which colours are in the code at all. Once repeats are allowed, a guess with two pairs, such as two reds and two blues, is a classic opener because it narrows the possibilities well.

Count colours before positions

Add the filled discs and hollow rings together: that total is how many of your colours are in the code, regardless of position. Work out which colours belong first, then use the filled discs to settle where each one goes.

Treat an empty hole like a colour

From level 13 a hole may be empty, and an empty hole scores markers exactly like a colour. If no colour seems to account for the last missing peg, try an empty hole in that spot.

Keep track of what is ruled out

A guess with no markers at all is very useful: every colour in it is out of the code. On the per-peg levels, a cross under a peg tells you the same thing about that colour.

The science behind the Deductive Reasoning

Mastermind, the code-breaking game invented by Mordecai Meirowitz in 1970, is an exercise in deductive reasoning: each guess is a hypothesis about the hidden code, and its markers rule possibilities in or out. Donald Knuth (1977) proved that any code of four pegs in six colours can be cracked in five guesses or fewer. Patrick Laughlin and colleagues (1982) studied it as a concept-learning problem and described two ways of choosing guesses, focusing and tactical, and John Best (1990) found that players learn to make guesses whose feedback is easy to interpret. The best guess here is one that fits every clue so far.

Frequently asked questions

What does a code-breaking game measure?

Code-breaking games exercise deductive reasoning and hypothesis testing: forming a guess about a hidden answer, testing it, and combining the results of several tests to rule possibilities in or out. Researchers have used them to study how people choose informative tests and how their strategies improve with practice.

What is the best strategy for cracking a colour code?

Make each guess consistent with all the clues so far, and when the clues are hard to read, change only a peg or two at a time so that any change in the markers is easy to explain. Donald Knuth showed that with perfect play the classic four-peg, six-colour code never needs more than five guesses.

What do the black and white pegs mean in the board game?

In the classic board game, a black peg means a peg in your guess is the right colour in the right position, and a white peg means it is the right colour in the wrong position. In Neuroptimize these are a filled disc and a hollow ring, and the per-peg levels add a cross for a colour that is not in the code.

Is Deductive Reasoning free?

Yes, it is a free game. It unlocks once your Neuroptimize account reaches level 14, and Pro members can play it straight away.