The graphics used in the previous posts in this blog have been made using the Sudoku Instructions program, which we have developed. You can obtain it from Sudoku Instructions website. The program is very user-friendly. It is designed to assist you in solving any sudoku puzzle by guiding you all the way through the solving process.

We wish you the best of luck in your Sudoku playing! Best regards, ERVICH

## Monday, November 9, 2009

## Saturday, November 7, 2009

### Sudoku: ‘X-Wing’, ‘Swordfish’ and ‘Jellyfish’ Clarified

These more difficult methods for candidate reduction in Sudoku are based on the relationship of a candidate number with both rows and columns. The X-wing is applied for two rows and two columns, the Swordfish for three rows and three columns and the Jellyfish for four rows and four columns. For an X-wing to be present the same candidate number must be present in all four squares where the rows and columns cross each other. For swordfish and jellyfish the candidate number in question need not be present in all the squares where the columns and rows cross each other. The rather odd names of these methods may relate very faintly to the outline of the involved squares with the candidate number in question. We will understand more of these difficult methods if we start with the simplest, namely the X-wing.

If the same candidate number within two rows is limited to the same two columns, then that number must necessarily be in two of the four ‘crossing-squares’ where the rows and columns cross each other. Since that number can only be once in each of the two columns, the candidate number can be removed from the two columns outside the two rows. This method, which focuses on two times two squares, is called an X-wing. We do not know in which two of the four ‘crossing-squares’ the number should be placed. Either the number should be placed as indicated by the \ part of the X or as indicated by the / part of the X. The exact place would become clear later in the solving process.

This board shows an 'X-wing' for candidate number 8 (shown in blue), which forms an X-wing in squares G2, G4, I2 and I4, because in rows 2 and 4 candidate number 8 is only present in columns G and I. So, in these columns candidate number 8 must be in rows 2 and 4. Therefore in these columns candidate number 8 can be removed from the other rows, here row 3 in squares G3 and I3 (shown in red).

In the small picture you can see the final solution for this part of the board. In row 2 number 8 should be in square G2 and in row 4 number 8 should be in square I4.

If the same candidate number within two columns is limited to the same two rows, then that number must necessarily be in two of the four ‘crossing-squares’ (the X-wing) where the columns and rows cross each other. Since that number can only be once in each of the two rows, the candidate number can be removed from the two rows outside the two columns. We do not know in which two of the four ‘crossing-squares’ the number should be placed. Either the number should be placed as indicated by the \ part of the X or as indicated by the / part of the X. The exact place would become clear later in the solving process.

This board shows an 'X-wing' for candidate number 5 (shown in blue), which forms an X-wing in squares H3, I3, H9 and I9, because in columns H and I candidate number 5 is only present in rows 3 and 9. So, in these rows candidate number 5 must be in columns H and I. Therefore in these rows candidate number 5 can be removed from the other columns, here columns D and F in squares D9 and F9 (shown in red).

In the small picture you can see the final solution for this part of the board. In column H number 5 should be in square H3 and in column I number 5 should be in square I9.

If the same candidate number within three rows is limited to the same three columns, then that number must necessarily be in three of the ‘crossing-squares’ (the ‘swordfish’) where the rows and columns cross each other. Since that number can only be once in each of the three columns, the candidate number can be removed from the three columns outside the three rows. We do not know in which three of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Swordfish' for candidate number 2 (shown in blue), which forms a swordfish in squares A5, A8, A9, E5, E8, E9, F5 and F8, because in rows 5, 8 and 9 candidate number 2 is only present in columns A, E and F. So, in these columns candidate number 2 must be in rows 5, 8 and 9. Therefore in these columns candidate number 2 can be removed from the other rows, here row 4 in squares A4 and F4 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 2 should in square A9 in column A, in square E8 in column E and in square F5 in column F.

If the same candidate number within three columns is limited to the same three rows, then that number must necessarily be in three of the ‘crossing-squares’ (the ‘swordfish’) where the columns and rows cross each other. Since that number can only be once in each of the three rows, the candidate number can be removed from the three rows outside the three columns. We do not know in which three of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Swordfish' for candidate number 8 (shown in blue), which forms a swordfish in squares A2, A4, C2, C5, E4 and E5, because in columns A, C and E candidate number 8 is only present in rows 2, 4 and 5. So, in these rows candidate number 8 must be in columns A, C and E. Therefore in these rows candidate number 8 can be removed from the other columns, here column F in squares F4 and F5 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 8 should in square A2 in column A, in square C5 in column C and in square E4 in column E.

If the same candidate number within four rows is limited to the same four columns, then that number must necessarily be in four of the ‘crossing-squares’ (the ‘jellyfish’) where the rows and columns cross each other. Since that number can only be once in each of the four columns, the candidate number can be removed from the four columns outside the four rows. We do not know in which four of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Jellyfish' for candidate number 2 (shown in blue), which forms a jellyfish in squares D3, D5, E2, E3, E5, G2, G3, G4, H2, H4 and H5, because in rows 2, 3, 4 and 5 candidate number 2 is only present in columns D, E, G and H. So, in these columns candidate number 2 must be in rows 2, 3, 4 and 5. Therefore in these columns candidate number 2 can be removed from the other rows, here row 1 in squares D1, E1 and H1 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 2 should in square D5 in column D, in square E2 in column E, in square G3 in column G and in square H4 in column H.

If the same candidate number within four columns is limited to the same four rows, then that number must necessarily be in four of the ‘crossing-squares’ (the ‘jellyfish’) where the columns and rows cross each other. Since that number can only be once in each of the four rows, the candidate number can be removed from the four rows outside the four columns. We do not know in which four of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Jellyfish' for candidate number 8 (shown in blue), which forms a jellyfish in squares D2, D3, D8, F2, F7, F8, H2, H3, H7, H8, I2, I7 and I8, because in columns D, F, H, and I candidate number 8 is only present in rows 2, 3, 7 and 8. So, in these rows candidate number 8 must be in columns D, F, H and I. Therefore in these rows candidate number 8 can be removed from the other columns, here column E in squares E2, E3 and E7 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 8 should in square D3 in column D, in square F8 in column F, in square H7 in column H and in square I2 in column I.

As you have seen these methods are not very easy to use. They need access to the candidate table and even then it may be difficult to spot a jellyfish or a swordfish. The X-wing, which is the simplest of these methods, may also be rather difficult to spot. To apply these methods in practice you would have great help of a user-friendly Sudoku program.

**Sudoku X-wing – Rows/Columns**If the same candidate number within two rows is limited to the same two columns, then that number must necessarily be in two of the four ‘crossing-squares’ where the rows and columns cross each other. Since that number can only be once in each of the two columns, the candidate number can be removed from the two columns outside the two rows. This method, which focuses on two times two squares, is called an X-wing. We do not know in which two of the four ‘crossing-squares’ the number should be placed. Either the number should be placed as indicated by the \ part of the X or as indicated by the / part of the X. The exact place would become clear later in the solving process.

This board shows an 'X-wing' for candidate number 8 (shown in blue), which forms an X-wing in squares G2, G4, I2 and I4, because in rows 2 and 4 candidate number 8 is only present in columns G and I. So, in these columns candidate number 8 must be in rows 2 and 4. Therefore in these columns candidate number 8 can be removed from the other rows, here row 3 in squares G3 and I3 (shown in red).

In the small picture you can see the final solution for this part of the board. In row 2 number 8 should be in square G2 and in row 4 number 8 should be in square I4.

**Sudoku X-wing – Columns/Rows**If the same candidate number within two columns is limited to the same two rows, then that number must necessarily be in two of the four ‘crossing-squares’ (the X-wing) where the columns and rows cross each other. Since that number can only be once in each of the two rows, the candidate number can be removed from the two rows outside the two columns. We do not know in which two of the four ‘crossing-squares’ the number should be placed. Either the number should be placed as indicated by the \ part of the X or as indicated by the / part of the X. The exact place would become clear later in the solving process.

This board shows an 'X-wing' for candidate number 5 (shown in blue), which forms an X-wing in squares H3, I3, H9 and I9, because in columns H and I candidate number 5 is only present in rows 3 and 9. So, in these rows candidate number 5 must be in columns H and I. Therefore in these rows candidate number 5 can be removed from the other columns, here columns D and F in squares D9 and F9 (shown in red).

In the small picture you can see the final solution for this part of the board. In column H number 5 should be in square H3 and in column I number 5 should be in square I9.

**Sudoku Swordfish - Rows/Columns**If the same candidate number within three rows is limited to the same three columns, then that number must necessarily be in three of the ‘crossing-squares’ (the ‘swordfish’) where the rows and columns cross each other. Since that number can only be once in each of the three columns, the candidate number can be removed from the three columns outside the three rows. We do not know in which three of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Swordfish' for candidate number 2 (shown in blue), which forms a swordfish in squares A5, A8, A9, E5, E8, E9, F5 and F8, because in rows 5, 8 and 9 candidate number 2 is only present in columns A, E and F. So, in these columns candidate number 2 must be in rows 5, 8 and 9. Therefore in these columns candidate number 2 can be removed from the other rows, here row 4 in squares A4 and F4 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 2 should in square A9 in column A, in square E8 in column E and in square F5 in column F.

**Sudoku Swordfish – Columns/Rows**If the same candidate number within three columns is limited to the same three rows, then that number must necessarily be in three of the ‘crossing-squares’ (the ‘swordfish’) where the columns and rows cross each other. Since that number can only be once in each of the three rows, the candidate number can be removed from the three rows outside the three columns. We do not know in which three of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Swordfish' for candidate number 8 (shown in blue), which forms a swordfish in squares A2, A4, C2, C5, E4 and E5, because in columns A, C and E candidate number 8 is only present in rows 2, 4 and 5. So, in these rows candidate number 8 must be in columns A, C and E. Therefore in these rows candidate number 8 can be removed from the other columns, here column F in squares F4 and F5 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 8 should in square A2 in column A, in square C5 in column C and in square E4 in column E.

**Sudoku Jellyfish - Rows/Columns**If the same candidate number within four rows is limited to the same four columns, then that number must necessarily be in four of the ‘crossing-squares’ (the ‘jellyfish’) where the rows and columns cross each other. Since that number can only be once in each of the four columns, the candidate number can be removed from the four columns outside the four rows. We do not know in which four of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Jellyfish' for candidate number 2 (shown in blue), which forms a jellyfish in squares D3, D5, E2, E3, E5, G2, G3, G4, H2, H4 and H5, because in rows 2, 3, 4 and 5 candidate number 2 is only present in columns D, E, G and H. So, in these columns candidate number 2 must be in rows 2, 3, 4 and 5. Therefore in these columns candidate number 2 can be removed from the other rows, here row 1 in squares D1, E1 and H1 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 2 should in square D5 in column D, in square E2 in column E, in square G3 in column G and in square H4 in column H.

**Sudoku Jellyfish – Columns/Rows**If the same candidate number within four columns is limited to the same four rows, then that number must necessarily be in four of the ‘crossing-squares’ (the ‘jellyfish’) where the columns and rows cross each other. Since that number can only be once in each of the four rows, the candidate number can be removed from the four rows outside the four columns. We do not know in which four of the ‘crossing-squares’ the number should be placed, only that it cannot be in the same row or column. Their precise position would become clear later in the solving process.

This board shows a 'Jellyfish' for candidate number 8 (shown in blue), which forms a jellyfish in squares D2, D3, D8, F2, F7, F8, H2, H3, H7, H8, I2, I7 and I8, because in columns D, F, H, and I candidate number 8 is only present in rows 2, 3, 7 and 8. So, in these rows candidate number 8 must be in columns D, F, H and I. Therefore in these rows candidate number 8 can be removed from the other columns, here column E in squares E2, E3 and E7 (shown in red).

In the small picture you can see the final solution for this part of the board. Number 8 should in square D3 in column D, in square F8 in column F, in square H7 in column H and in square I2 in column I.

As you have seen these methods are not very easy to use. They need access to the candidate table and even then it may be difficult to spot a jellyfish or a swordfish. The X-wing, which is the simplest of these methods, may also be rather difficult to spot. To apply these methods in practice you would have great help of a user-friendly Sudoku program.

## Thursday, November 5, 2009

### Sudoku: Hidden Candidates Reduction Method Clarified

In Sudoku ‘hidden’ candidates may be present in pairs, triples or quads. The candidates are ‘hidden’ if they are not alone but ‘hidden’ among other candidates in the squares. If ‘hidden’ candidates are present, those numbers must necessarily be in the squares where they are – they are linked to those squares. Therefore, any other candidates present in the squares with ‘hidden’ candidates can be removed from those squares. In the squares with ‘hidden’ candidates we do not yet know where each of the numbers should be placed, only that they collectively should be in those squares. Later in the solving process the unique position of each number will become clear. Now we will go through he method for pairs, triples and quads and give some examples.

If the same pair of candidates are ‘hidden’ among other candidates in just two squares within a group (row, column or block), then these candidates make up a ‘hidden’ pair. Neither of the candidates must be present in any other square within the group. Thus two numbers are limited and therefore linked to those two squares. Therefore, other candidate numbers can be removed from those two squares within the group.

This board shows a 'hidden' pair (1 and 8, shown in blue) in column A in squares A7 and A9. These candidates are 'hidden' because other candidates are also present in those squares. The numbers 1 and 8 are linked to those squares. Therefore the other candidate numbers, which are present in those squares, can be removed. The candidate numbers, which can be removed (6 and 9), are shown in red.

In the small picture you can see the final solution for this part of the board. In column A number 1 should be in square A7 and number 8 in square A9.

If the same three candidates are ‘hidden’ among other candidates in just three squares within a group (row, column or block), then these candidates make up ‘hidden’ triples. None of the candidates must be present in any other square within the group. Thus the three candidate numbers are limited to and therefore linked to those three squares. Therefore, other candidate numbers can be removed from those three squares within the group.

This board shows 'hidden' triples (1, 2 and 7, shown in blue) in row 3 in squares A3, C3 and F3. These candidates are 'hidden' because other candidates are also present in those squares. The numbers 1, 2 and 7 are linked to those squares. Therefore other candidate numbers being present in those squares can be removed. The candidate numbers, which can be removed (4 in A3, 6 in C3 and 6 and 9 in F3), are shown in red.

In the small picture you can see the final solution for this part of the board. In row 3 number 1 should be in square A3, number 2 in square C3 and number 7 in square F3.

If the same four of candidates are ‘hidden’ among other candidates in just four squares within a group (row, column or block), then these candidates make up ‘hidden’ quads. None of the candidates must be present in any other square within the group. The three numbers are limited to and therefore linked to those four squares. Therefore, other candidate numbers can be removed from those four squares within the group.

This board shows 'hidden' quads (2, 3, 4 and 6, shown in blue) in block P in squares B2, C1, C2 and C3. These candidates are 'hidden' because other candidates are also present in those squares. The numbers 2, 3, 4 and 6 are linked to those squares. Therefore other candidate numbers being present in those squares can be removed. The candidate numbers, which can be removed (5 from C1, 7 from B2, C2 and C3 and 8 from B2, C1, C2 and C3), are shown in red.

In the small picture you can see the final solution for this part of the board. In block P number 3 should be in square B2, number 6 in square C1, number 4 in square C2 and number 2 in square C3.

This method of finding ‘hidden’ candidates in a Sudoku puzzle can be used repeatedly for candidate reduction. Thereby, more single candidates may appear making it easier to solve the Sudoku puzzle. In the next post we will go through even more elaborate methods of candidate reduction.

**Sudoku Hidden Pair**If the same pair of candidates are ‘hidden’ among other candidates in just two squares within a group (row, column or block), then these candidates make up a ‘hidden’ pair. Neither of the candidates must be present in any other square within the group. Thus two numbers are limited and therefore linked to those two squares. Therefore, other candidate numbers can be removed from those two squares within the group.

This board shows a 'hidden' pair (1 and 8, shown in blue) in column A in squares A7 and A9. These candidates are 'hidden' because other candidates are also present in those squares. The numbers 1 and 8 are linked to those squares. Therefore the other candidate numbers, which are present in those squares, can be removed. The candidate numbers, which can be removed (6 and 9), are shown in red.

In the small picture you can see the final solution for this part of the board. In column A number 1 should be in square A7 and number 8 in square A9.

**Sudoku Hidden Triples**If the same three candidates are ‘hidden’ among other candidates in just three squares within a group (row, column or block), then these candidates make up ‘hidden’ triples. None of the candidates must be present in any other square within the group. Thus the three candidate numbers are limited to and therefore linked to those three squares. Therefore, other candidate numbers can be removed from those three squares within the group.

This board shows 'hidden' triples (1, 2 and 7, shown in blue) in row 3 in squares A3, C3 and F3. These candidates are 'hidden' because other candidates are also present in those squares. The numbers 1, 2 and 7 are linked to those squares. Therefore other candidate numbers being present in those squares can be removed. The candidate numbers, which can be removed (4 in A3, 6 in C3 and 6 and 9 in F3), are shown in red.

In the small picture you can see the final solution for this part of the board. In row 3 number 1 should be in square A3, number 2 in square C3 and number 7 in square F3.

**Sudoku Hidden Quads**If the same four of candidates are ‘hidden’ among other candidates in just four squares within a group (row, column or block), then these candidates make up ‘hidden’ quads. None of the candidates must be present in any other square within the group. The three numbers are limited to and therefore linked to those four squares. Therefore, other candidate numbers can be removed from those four squares within the group.

This board shows 'hidden' quads (2, 3, 4 and 6, shown in blue) in block P in squares B2, C1, C2 and C3. These candidates are 'hidden' because other candidates are also present in those squares. The numbers 2, 3, 4 and 6 are linked to those squares. Therefore other candidate numbers being present in those squares can be removed. The candidate numbers, which can be removed (5 from C1, 7 from B2, C2 and C3 and 8 from B2, C1, C2 and C3), are shown in red.

In the small picture you can see the final solution for this part of the board. In block P number 3 should be in square B2, number 6 in square C1, number 4 in square C2 and number 2 in square C3.

This method of finding ‘hidden’ candidates in a Sudoku puzzle can be used repeatedly for candidate reduction. Thereby, more single candidates may appear making it easier to solve the Sudoku puzzle. In the next post we will go through even more elaborate methods of candidate reduction.

Subscribe to:
Posts (Atom)