Sudoku - The Difficult Puzzles

Solution with Hidden Number Twice Removed

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 second row there is a seven which prevents a seven being put anywhere else in that row:

In the third row there is also a seven, so that row is also blocked to sevens:

There must be a seven in field 2, so the only place to put it is centre-top:

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 7 had been reached. Scrutiny of the puzzle will show that the eights and nines cannot be "chased" either.

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 7 has been put in one should check the numbers 1 to 6 again.

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 1. Look at the extreme top left square. It has the numbers 3, 7 and 9 in its row. It has the numbers 6 and 8 in its column. It has also a 6 and 7 in its field. That makes seven entries, but we need eight. In addition, they must be unique. The 6 in the field duplicates that in the column. The seven in the field duplicates that in the row. This gives an incomplete analysis of 3r 6cf 7rf 8c 9r. Here, r is "row", c is "column" and f is "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 five unique numbers in seven separate entries. Insufficient to justify analysis.

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

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

Move 2

We put it in:

Now that the eight is there, we can chase another eight into position:

Move 3

We put it in:

That new eight enables yet one more to be chased into position:

Move 4

We put it in:

The seven that we started with also lets us chase a 6 into position:

Move 5

Which we also put in:

An analysis of the second row gives 1m 2c 3c 4r 5r 6r 7rc 8rc 9r:

Move 6

We put in the missing 1:

Now we chase the 5:

Move 7

We put in that 5:

That five helps us to analyse the second square:

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

Move 8

We put in the 4:

That enables us to chase another 4:

Move 9

into position:

That 4 lets us chase yet another:

Move 10

and here it is:

The "Hidden Numbers"

In the special case of Move 11, we can use hidden numbers:

Here, the 3 on the top forces the 3 in the third field to be in the ninth column. The position is ambiguous. That is why there are two red threes. However, whichever is correct, it will always block off the ninth column to threes, and so helps us chase a three into field 9.

An alternative way of looking at the seventh column is to observe that all other vacant places are obstructed to the 3. The topmost vacant place is obstructed by the 3 in field two. Three places below that are obstructed by the 3 in field six. Below those, a square is obstructed by a 3 in field seven. That leaves only the bottom-most place to put a 3:

Move 11

Here is that 3 in position:

"Hidden Numbers Twice Removed"

A 9 in field three forces a hidden 9 to be in row five:

That is the first stage of removal. That obstructed row forces a 9 to be hidden in column three:

That is the second stage of removal. It helps us chase a 9 into field one:

Move 12

So we put in that 9:

That 9 disambiguates the position of the 5 in field one:

Move 13

So the 5 goes in:

"Exclusion"

Row three needs 1, 2, 8 and 3. However, the 1, 2 and 8 are needed for field two. So they are excluded from the final position in the row. Similarly, there is already a 3 in field two. The new 3 is excluded from the middle, and banished to the end of the row:

Move 14

The 3 is put in:

This enables another analysis:

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

Move 15

The 2 goes in:

Move 16

The 3 completes row two:

The previous 2 lets us chase another:

Move 17

into field nine:

which lets us chase another:

Move 18

into field seven:

which lets us chase another:

Move 19

into field one:

Move 20

The 1 completes field one:

which lets us chase a 1:

Move 21

into field seven:

and a 1 again:

Move 22

into field nine:

which disambiguates the position of the 9:

Move 23

in row eight:

Move 24

where a 7 completes the row:

and lets us chase another 7:

Move 25

into field four:

which disambiguates the position of the 5:

Move 26

which can be put into field four:

Move 27

The 4 completes the first column:

Move 28

and the 5 completes the second column:

A 7 on the bottom row is only possible:

Move 29

in the extreme bottom right of the puzzle:

A 6 on the bottom row is only possible:

Move 30

beside that 7:

Move 31

and a 9 completes the bottom row:

Move 32

another 9 completes field nine:

Move 33

Yet another 9 completes column eight:

5, 6 and 7 are required in row seven, but 6 and 7 are excluded from the middle of the puzzle:

Move 34

so the 5 goes in:

This five lets us chase the first thing into the middle field:

Move 35

It is a 5:

An analysis to the immediate left of it gives 1r 2r 3m 4r 5rc 6c 7c 8r 9c

Move 36

So the 3 goes in:

Column 3 needs 3, 4 and 9. The new 3 and a 4 reduce the options to the 9:

Move 37

So the 9 goes in:

A 3 can only go above the 9:

Move 38

So we put in a 3:

Move 39

and a 4 completes the column:

The 4s define the centre of the puzzle:

Move 40

to be a 4:

The central column has only two places for a 2, one of them wrong:

Move 41

so a 2 goes in:

Move 42

and a 1 completes the column:

which lets us chase a 1 into field six:

Move 43

as shown:

The centre is sufficiently full:

Move 44

to chase a 2 into position:

Column six:

Move 45

needs a 6:

and a 7:

Move 46

as shown:

Move 47

and an 8 completes that column:

Move 48

A 1 completes field two:

A 9 can be chased into field five:

Move 49

as here:

Move 50

and an 8 completes row four:

Column seven:

Move 51

acquires a 7:

Move 52

and a 6 completes the column:

Move 53

An 8 completes field three:

Move 54

A 6 completes column nine:

Move 55

An 8 completes row five:

Move 56

A 6 completes row six:

Move 57

A 7 completes row seven:

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.

In addition, there is human error. Observe, for example, Move 7. This move could have been made right at the start, but was overlooked:

However, the mindset whilst solving this puzzle was to use many different techniques at the beginning, so as to get the descriptions dealt with. The latter part of the solution involved the use of techniques previously described.

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