# Noughts, Crosses, Question marks, and Ouroboros

In the following diagram, identify the symbols that go to the position marked as ****. Each arrow implies a specific relationship between its head and its tail.

+------> OXOO -------+
|                    |
|                    v
O?O?              ?XXO
^                    |
|                    |
|   **** ---> OOOO   |
|                    |
|                    |
+-- X??O <--- ?X?? <-+


Hint 1:

A question mark (?) is just a symbol like an O or an X. It isn't something that hides another symbol.

Hint 2:

Commonly the symbol O means correct, and X means incorrect. The question mark means somewhere in between. But these meanings apply to only one (consistent) end of each arrow.

X?O? -> OOOO

Method:

Each set of 4 symbols represents

A guess in a game of Mastermind. The opponent has a hidden sequence of 4 symbols.

Following round the arrows, the next set of 4 symbols represents

The response from the opponent.

Specifically:

O represents a guessed symbol that was in the right place
X represents a guessed symbol that does not appear
? represents a guessed symbol that does appear, but not in the location stated
The goal of the game is to guess the correct set of symbols in the correct order, i.e. to get a response of OOOO

Working through the individual clues:

1. OXOO -> ?XXO tells us

there are either exactly two O and no X in the final solution, or exactly one O and at least one X. Exactly one of the correct symbols is in the right place.

1. ?XXO -> ?X?? tells us

either all 3 symbols are in the final solution with exactly one X, or there are no ? and at least two X

1. ?X?? -> X??O tells us

either there are 3 ? in the final solution, or exactly two ? and at least one X. Taken with the preceding two clues, the final solution would appear to be a permutation of ??XO, with one of the ? in position 2 (as one was in the wrong place, and this is the only position that a ? was not guessed)

1. X??O -> O?O? tells us

the previous conclusion was indeed correct. Two symbols are in the right place, the other two must be swapped, and we know position 2 is correct. Permutations fitting this with 1 swap: ??XO violates clue 2, O??X or X?O? work so far.

1. O?O? -> OXOO tells us

The final solution must look like *?O? or O?*?. Only one of the permutations that had worked for the previous clues matches this pattern.

Rechecking what we learned so far

X?O? seems to be the correct pattern, so
OXOO -> ?XXO (X in wrong place, one O in right place, no second or third O)
?XXO -> ?X?? (X, O, and ? all in wrong place, no second x)
?X?? -> X??O (final ? in right place, other ? and X in wrong place, no third ?)
X??O -> O?O? (X and one ? in right place, O and other ? in wrong place)
O?O? -> OXOO (*?O? is fully correct, but no second O)