In  the following problem is described:
- The chessboard shown above has \(8\times 8 = 64\) cells.
- Place numbers 1 through 64 in the cells such that the next number is either directly left, right, below, or above the current number.
- The dark cells must be occupied by prime numbers. Note there are 18 dark cells, and 18 primes in the sequence \(1,\dots,64\). These prime numbers are: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61.
This problem looks very much like the Numbrix puzzle . We don't have the given cells that are provided in Numbrix, but instead, we have some cells that need to be covered by prime numbers.
The path formed by the consecutive values looks like how a rook moves on a chessboard. Which explains the title of this post (and of the original post). If we require a proper tour we need values 1 and 64 to be neighbors. Otherwise, if this is not required, I think a better term is: Hamiltonian path. I'll discuss both cases, starting with the Hamiltonian path problem.