# The Sudoku Solution based on the Chain of Head and Tail Extremes.

The Chain of Head and Tail Extremes.

As shown in the board, the chain is written as follows:

H1(1) = A1(1-8) = A6(8) – I6(8) = I4(8-2) = H4(2) => H1<>2, H4<>1

At this point, we find that the chain head and tail are strange: they are not the same candidate numbers. What is the logic of deleting numbers in this chain? Why shouldn’t we fill in 2 for H1 and 1 for H4?

This is because if we start reasoning according to the initial “assumption,” we have the following statements:

- Step 1: H1<>1, since column 1 must have a 1, then A1=1.
- Step 2: A1=1, so A1<>8.
- Step 3: A1<>8, since row A must have an 8, then A6=8.
- Step 4: A6=8, since column 6 cannot have the 2nd 8, then I6<>8.
- Step 5: I6<>8, since row I must have an 8, then I4=8.
- Step 6: I4=8, so I4<>2.
- Step 7: I4<>2, since column 4 must have a 2, then H4=2.

Now we find that when H1=1, how to delete the candidates?

As mentioned earlier, no matter which scenario, the chain head and tail will only have one number to be filled. Therefore, we need to delete the candidate number that corresponds to the common position of the chain head and tail. In the cell where the chain head is located, the candidate number that is the same as the chain tail can be deleted. Similarly, in the cell where the chain tail is located, the candidate number that is the same as the chain head can also be deleted. The candidate number 2 of H1 and the candidate number 1 of H4 happen to correspond to the chain head and tail, so they can be deleted. Why can this be considered a correspondence? Assume that when H1=1, H1<>2 (since H1 has already been filled in with a number, all other candidate numbers are wrong), and H4<>1 (because H1 and H4 are in the same row H, only one 1 can appear). When H4=2, H4 cannot fill in 1 naturally, and H1 cannot fill in 2. Both scenarios can lead to H1 not filling in 2 and H4 not filling in 1.

This technique is called a “head and tail extremes chain.” It is easy to understand because the head and tail of the chain are in the same cell.