Here the 3 cases are:
1. all the same
2. single outlier in the middle
3. single outlier at the border
This should be extendable to 64 squares, but the pattern rules will become more and more complicated.
Dyebukk(YT)
17 July 2026
The optimal strategy uses binary XOR (Exclusive OR) logic to encode the target square's position into the board's state.
The Core Strategy should be the prisoners' number the chessboard squares from 0 to 63. They treat the board as a single binary calculation where flipping one coin changes the mathematical "checksum" of the board to match the warden's chosen square.
Execution
Number the board: Assign every square a number from 0 to 63 (expressed in 6-bit binary).
Define coin states: Value Tails as 0 and Heads as 1.
Calculate initial state: Prisoner A XORs the numbers of all squares currently showing Heads.
Determine the flip: Prisoner A XORs this initial board value with the warden's target square number.
Flip the coin: The resulting number is the exact square Prisoner A must flip.
Read the board: Prisoner B enters and XORs all squares showing Heads.
Name the square: The final XOR sum perfectly equals the warden's target square.
For Example:
Flipping a coin at position N alters the total XOR sum of the board by exactly N. If the initial board XOR sum is 10 and the warden picks square 42:
Prisoner A calculates 10 oplus 42 = 32.
Prisoner A flips the coin on square 32.
The new board XOR sum becomes 10 oplus 32 = 42.
Prisoner B counts the Heads, gets 42, and wins.
If the calculation yields 0, Prisoner A flips square 0, which changes the coin but mathematically alters the XOR sum by 0.
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Comments (3)
Does A turn the coin (A chooses) over or flip it by throwing it (random)?
Solution for a 2-square chessboard:
HH -> HH / TH
HT -> HH / HT
TH -> HH / TH
TT -> TT / HT
The 2 cases are:
1. both the same
2. different
And now a solution for a 3-square chessboard:
HHH -> HHH / HTH / HHT
HHT -> HHH / THT / HHT
HTH -> HHH / HTH / TTH
HTT -> TTT / HTH / HHT
THH -> HHH / THT / THH
THT -> TTT / THT / THH
TTH -> TTT / HTH / THH
TTT -> TTT / THT / TTH
Here the 3 cases are:
1. all the same
2. single outlier in the middle
3. single outlier at the border
This should be extendable to 64 squares, but the pattern rules will become more and more complicated.
The optimal strategy uses binary XOR (Exclusive OR) logic to encode the target square's position into the board's state.
The Core Strategy should be the prisoners' number the chessboard squares from 0 to 63. They treat the board as a single binary calculation where flipping one coin changes the mathematical "checksum" of the board to match the warden's chosen square.
Execution
Number the board: Assign every square a number from 0 to 63 (expressed in 6-bit binary).
Define coin states: Value Tails as 0 and Heads as 1.
Calculate initial state: Prisoner A XORs the numbers of all squares currently showing Heads.
Determine the flip: Prisoner A XORs this initial board value with the warden's target square number.
Flip the coin: The resulting number is the exact square Prisoner A must flip.
Read the board: Prisoner B enters and XORs all squares showing Heads.
Name the square: The final XOR sum perfectly equals the warden's target square.
For Example:
Flipping a coin at position N alters the total XOR sum of the board by exactly N. If the initial board XOR sum is 10 and the warden picks square 42:
Prisoner A calculates 10 oplus 42 = 32.
Prisoner A flips the coin on square 32.
The new board XOR sum becomes 10 oplus 32 = 42.
Prisoner B counts the Heads, gets 42, and wins.
If the calculation yields 0, Prisoner A flips square 0, which changes the coin but mathematically alters the XOR sum by 0.
Add a Comment or Suggest an Answer