This colorful twist on a logic puzzle goes as follows:

Seven prisoners, each wearing a rainbow-colored hat (chosen from 7 possible colors) stand in a line or act simultaneously.
Each sees the other six hats, but not their own.
Without communication after the hats are placed, each prisoner must guess the color of their own hat.

If at least one guess is correct, all go free.
If no one guesses correctly, they are executed.

Can the prisoners devise a strategy that guarantees survival?

Step 1: Key Observations

  • There are 7 possible hat colors (think: Red, Orange, Yellow, Green, Blue, Indigo, Violet).
  • Hats may repeat.
  • The challenge is to ensure at least one prisoner guesses correctly—every time.

Step 2: Use Modulo Arithmetic (Z₇)

Label the hat colors with numbers:

\[ \text{Red} = 0,\quad \text{Orange} = 1,\quad \ldots,\quad \text{Violet} = 6 \]

Now think of the color assignments as values in mod 7 arithmetic.

Let the prisoners be numbered \(P_0, P_1, \ldots, P_6\).

Each prisoner can see the hat colors of everyone else, so they know the values \(c_0, \ldots, c_6\) except for their own.

Step 3: Strategy

The prisoners agree on the following rule:

Each prisoner \(P_i\) assumes that the sum of all hat colors modulo 7 is congruent to \(i\).
Then prisoner \(P_i\) guesses their own hat color as:

\[ g_i = (i - \sum_{j \ne i} c_j) \mod 7 \]

So each prisoner computes what their own hat color must be to make the total modulo 7 sum equal to their index.

Step 4: Why It Works

Let’s suppose the actual hat colors are \(c_0, c_1, \ldots, c_6\).

Then the true total sum is:

\[ T = \sum_{i=0}^{6} c_i \mod 7 \]

Now, only the prisoner \(P_T\) will have guessed their own hat color correctly:

  • \(P_T\) assumes the total should be \(T\), and since the real total is \(T\), their calculated guess is correct.

All other prisoners will be wrong—but one will always be right.

Final Answer

Yes, the prisoners can guarantee freedom by using a mod 7 parity strategy.
Each assumes the total color sum mod 7 equals their index, and deduces their own color accordingly.
Exactly one prisoner is guaranteed to guess correctly—ensuring they all survive.

Reference