Naked Quad

When four cells that see each other have only four digits as candidates, the digits are called a Naked Quad. We can eliminate the other occurrences of these digits from all the cells that see all four cells of the Naked Quad. The reason for the elimination is that the solution for these cells must be the Naked Quad digits, which means no cell that sees all the cells of the Naked Quad can contain any of these digits.

In classic Sudoku, all the cells of the Naked Quad are in the same region (same row, column, or box), and all the cells that see the Naked Quad cells must be in the same Region.

Example 1

In the image below, in column 1, digits 4, 5, 7 & 8 in cells B1, F1, H1, and I1 form a Naked Quad. These cells don’t have any other digits as candidates, and their solution must be 4, 5, 7, & 8. Therefore, the occurrences of 4, 5, 7 & 8 can be eliminated from other cells in column 1 since those cells see all the cells of the Naked Quad.

Example 2

In row E, the digits 3, 6, 7 & 8 in cells E1, E2, E3 & E4 form a Naked Quad. The solution for E1, E2, E3 & E4 must be the digits 3, 6, 7 & 8. Therefore, 3, 6, 7 & 8 can be eliminated from the rest of the cells in row E.

Example 3

In this example, we are extending the Naked Quad into the variants world. The puzzle here includes, in addition to classic Sudoku, the killer variant. The shaded cells in box 6 form a Naked Quad with the digits 2, 4, 6, 8. The cell G7 is in the same cage as the Naked Quad cells. Digits in a cage cannot repeat, so we can eliminate the digits 2, 4, 6 & 8 from G7.

Example 4

This is another variants example of Naked Quad. The puzzle below contains the Chess King and Chess Knight variants, in addition to classic Sudoku. The digits 1, 2, 5 & 9 form a Naked Quad in cells F2, G3, G4, and H3. Notice that each cell sees the other three cells by the rules of one of the variants or classic Sudoku. Since all four cells see each other, their solution must be the digits 1, 2, 5 & 9. As a result, we can eliminate 1, 2, 5 & 9 from all the cells that see all the cells of the Naked Quad. In this example, 1, 2, 5 & 9 can be eliminated from G1 since it sees the other four cells. It sees F2 because of Chess King rules and sees G3, G4 & H3 because of classic Sudoku rules.

Example 5

Another example of Naked Quad with variants. The puzzle here includes the Diagonal and Non-consecutive variants in addition to classic Sudoku rules. The important variant here is Diagonal, as it is a region that must include all the digits 1-9 only once. The digits 1, 6, 8 & 9 in cells A9, B8, H2, and I1 form a Naked Quad on the diagonal that goes from top right to bottom left. (It is true that the pairs 8 & 9 and 1 & 6 are also Naked Pairs). The solution for those cells must be those digits, and those digits cannot be the solution for any other cell on this diagonal. Therefore, 1, 6, 8 & 9 can be eliminated from other cells on this diagonal.