Sudoku - The Difficult Puzzles

Solution with Two Hidden Numbers

Here we have a typical puzzle.

 

WE will define some of the properties of a Sudoku. The whole thing will be called a "puzzle". A series of nine squares horizontally will be called a "row". We number the rows from 1 to 9. Row 1 is at the top. A series of nine squares above each other will be called a "column". We number these also from 1 to 9. Column 1 is on the left. A set of nine squares, three rows by three columns, bounded by a bold line, will be defined as a "field". We count the fields from the top left to the bottom right. Thus: 1,2,3 and in the middle 4,5,6 and at the bottom 7,8,9. We also count the squares in a field that way.

The "Chase"

Examining the puzzle, we note that in the eighth row there is a 5 which prevents a 5 being put anywhere else in that row:

 

In the ninth row there is also a 5, so that row is also blocked to fives:

 

In the seventh column there is also a 5, so that column is also blocked to fives:

 

There must be a 5 in field nine, so the only place to put it is centre-top of that field:

 

Move 1

This, therefore, is the first move:

 

The "Chase" is a quick and effective way of filling a puzzle. However, it is used more on the easy puzzles. In this example, attempts were made to "chase" the ones, then the two and so on. It only became possible to chase a number into position when 5 had been reached.

If one attempts to chase all numbers from 1 to 9, and fails, and has not overlooked anything, it becomes pointless to start again at 1. However, if a number has been put in, it may alter things. For example, if a 5 has been put in one should check the numbers 1 to 4 again.

We cannot chase a 6, 7 or 8. However we can chase a 9:

 

Move 2

into the seventh field:

 

The "Analysis"

A row, column or field must contain all nine numbers, 1 to 9. However, as each square sits in a row, a column and a field, it is possible for the eight other numbers to be shared amongst the row, column and field.

Let us do some pre-analysis.

Consider the puzzle after Move 2. Look at the extreme top left square. It has the numbers 2 and 5 in its row. It has the numbers 9 and 7 in its column. That makes four entries, but we need eight. In addition, they must be unique. This gives an incomplete analysis of 2r 5r 7c 9c. Here, r is "row", c is "column" and f would be "field". If the same entry is in a row and field, or in a column and field, the f is not quoted. Thus there can be seen to be four unique numbers in four separate entries. Insufficient to justify analysis.

There is another position, however, where things are different:

 

The analysis here is 1c 2m 3c 4c 5c 6r 7r 8r 9r, where 2m means 2 is missing.

Move 3

We put it in:

 

A further analysis at the end of row six is 1r 2r 3r 4c 5m 6f 7r 8f 9c.

 

Move 4

We put a 5 in:

 

Again, we have 1m 2r 3f 4c 5c 6r 7rf 8r 9rc at the end of row four.

 

Move 5

The 1 goes in:

 

That 1 lets us chase another into the centre of the puzzle.

 

Move 6

Here it is:

 

At the end of the middle row, we can do an analysis, giving 1rc 2m 3f 4c 5c 6f 7f 8f 9c:

 

Move 7

Where the 2 goes in:

 

Double "Hidden Numbers"

In row four the 3, 4 and 5 are missing (as shown small).

On the top row, the 5 prevents field two having a 5 on the top. This means, it must be one of the red 5s in the middle column. This prevents the top centre of field five having a 5. The 5 is shown in blue.

On the bottom row, the 3 prevents another 3 being at the bottom. So it is one of the 3s in the centre column, which obstructs row four from having a 3. The 3 is shown in blue.

The remaining option, the 4, is shown in red:

 

Move 8

So the 4 goes in:

 

That 4 lets us chase another

 

Move 9

into position in field eight:

 

and another 4

 

Move 10

into the sixth row:

 

and another 4

 

Move 11

into the first field:

 

and another 4

 

Move 12

in the seventh field:

 

In the middle of the sixth row we need 6, 8 and 9. However, there is one position where the 6 and 8 are obstructed:

 

Move 13

leaving just the 9 for row six:

 

In row seven:

 

Move 14

there is just one place for a six:

 

then just one place for a 3:

 

Move 15

which is put in:

 

Move 16

and an 8 completes the row:

 

That 8 lets us chase another:

 

Move 17

into the top right of the puzzle:

 

In that last column, the 3:

 

Move 18

goes under the 8 in the top right:

 

In that last column, the ninth, the 6:

 

Move 19

is now obstructed everywhere except at the bottom:

 

Move 20

and a 7 completes column nine:

 

A recent 3:

 

Move 21

lets us chase another 3 to the top left:

 

and a 1:

 

Move 22

fits top left:

 

At this point, a 1 could have been put in:

 

or a 3:

 

but an analysis was spotted:

 

This is 1rc 2rc 3rf 4rcf 5r 6c 7m 8rc 9c.

Move 23

So the 7 goes in:

 

which lets us chase another 7:

 

Move 24

into the central field:

 

and another 7:

 

Move 25

to the bottom row:

 

The central bottom field needs only 8 or 9. Here, not the 9:

 

Move 26

so it is the 8:

 

Move 27

and the 9 completes the field:

 

There is an 8 in column four, not at the top:

 

Move 28

so the 8 is in row six:

 

Move 29

and it is the 6 at the top:

 

Move 30

a 6 completes row six:

 

Move 31

and a 9 completes the top row:

 

In row four, a 3 cannot be at the beginning:

 

Move 32

so the 3 is in the sixth column:

 

Move 33

with a 5 completing the fourth row:

 

Move 34

and a 5 completing both the central field and column six:

 

Here are two numbers that could have been chased:

 

Instead, an analysis was done:

 

It is 1c 2c 3rf 4rcf 5c 6m 7c 8r 9cf.

Move 35

so a 6 goes into column three:

 

underneath it, a 3:

 

Move 36

as shown:

 

Move 37

and an 8 completes column three:

 

a recent 6 in the first field:

 

Move 38

lets us chase a 6 into field three:

 

This new 6 in row three:

 

Move 39

sets us free to enter a 7:

 

Row three needs a 2 and a 5, but no 5 here:

 

Move 40

So it is a 2 in field three:

 

Move 41

and the 5 completes row three:

 

Move 42

The 2 completes field two:

 

There is only one place for a 5 in row two:

 

Move 43

The 5 is in the first field:

 

There is only one place for a 7 in row two:

 

Move 44

The 7 is in the third field:

 

Move 45

A 1 completes the first field:

 

Move 46

A 9 completes the second row and the third field:

 

There is only one place for a 6 in the first column:

 

Move 47

The 6 is in the fifth row:

 

There is only one place for an 8 in the first column:

 

Move 48

The 8 is bottom left:

 

Move 49

A 2 completes the first column:

 

A 3 in column two:

 

Move 50

belongs one place above the bottom:

 

Move 51

And a 1 completes field seven:

 

Move 52

whilst an 8 completes field four:

 

A 9 in column eight:

 

Move 53

goes into row five:

 

Move 54

A 4 completes field six:

 

Move 55

A 1 completes column seven:

 

Move 56

A 4 completes row eight:

 

Move 57

A 2 completes row nine:

 

and the puzzle is done!

 

The order in which the moves are made is influenced by the current move, but is not rigidly determined. A current move may open up a plurality of future moves. The order in which you make those moves is your decision.


BACK

(C) 2009 Charles Douglas Wehner.
Use freely but do not plagiarise.