Walberswick - June 2019

This year, Patrick and I took our summer break just before the solstice. As usual the Walberswick caravan site was almost empty of holiday makers. This suits our naturally anti-social characters admirably.

Our caravan towards the south western end of the site.

Monday, 17th

We tend to arrive a tad late on this first day. Bryony drives us and makes sure we get a decent meal; in this case chilli con carne and lots of it. She has a poor opinion of my cooking skills. Bryony then
snaffles the car and leaves P and I to fend for ourselves.

Organizing both ourselves and the caravan takes a while. Our ready made meal is very welcome. After this a postprandial walk is essential. The sun has dipped below the horizon but twilight makes seeing easy. Even so I took a torch, just in case. I headed to the beach. The tide was high, but the temperature was no more than moderately warm. I wasn't tempted in.

Walking along the top ridge of the dunes, I headed towards the track in land where we often find samphire in the standing pools. I couldn't spot any this year, but the light was fading.

Samphire

You can see from the map above, that there' is a footbridge over the Dunwich River. The path leads to a dyke between two drains. The drains eventually connect with the main river, leaving the walker the task of crossing them at the connecting point. Fortunately, this crossing was aided by a narrow (6") plank about 10 feet (3 m) long. This challenged my failing balance, but I made it with out losing my footing. 

The official path turns north, but this is both wet and difficult to see. Walkers have created a new passage along the the bank of the river. This has the advantage of being both higher and drier. Another path to the campsite still crosses the reed bed, but is much drier than the official version. Reed beds can be very tall. I was pleased to have my torch to light the way.

Tuesday, 18th

Caravan life is like camping. Your sleep patterns take a while to adjust. I woke shortly after dawn and slipped out for a sneaky piss in the open. Come to think of it, this has been a habit of mine since I was about 8 years old. After 60 years practice, I'm conversant with the precautions needed to stay unseen. Four or five years ago I might set off for my early morning walk immediately. As it was bed beckoned for another couple of hours.

Eventually, I rose, dressed and consumed my early morning "instant" around 06.15. (Bryony insists on the reference to "instant" to distinguish it from her real coffee.) You cannot really tell what walking conditions are going to be like, so walking boots are a necessity.

Just beyond the caravan site, a footbridge leads to the Walberswick summer campsite and beach huts.

Walberswick Campsite

At this time of year the campsite is empty. It has that serenity which marks its popularity during the six weeks of the school holidays. Although, its very popularity tends to undermine the feelings of serenity.

Walberswick's beach is mostly gravel. A short walk along it, no more than 300 yards, reminds me how heavy going beach walking is. It is a popular resort for those in the know. I spied one lone swimmer and a group of older swimmers in their bathrobes. As ever such groups are surrounded by a gaggle of dogs. Walberswick is one of the few resorts that doesn't ban dogs from the beach.

A cutting through the dunes gives a give view to the eccentric beach dwellings on this side of the Blyth River.

Holiday dwellings from the beach

 To get there I have to re-cross the Dunwich River which joins the Blyth just before spilling into the sea. This is rather aptly known as Wally's Bridge. There is a sign to dog owners to keep their dogs on a lead, since 8 have already fallen in the river. Perhaps the dogs should pull their owners in after them.

Through the beach dwellings, I came to the Walberswick ferry. It is still a rowing ferry. At that point I looked back into the village and spotted the Bell Inn; our favoured watering hole in Walberswick.

The Bell Inn from the harbour

The walk inland up the Blyth River is always a joy. Over 25 years ago, Rodney and I bought an open "day boat" together with an outboard motor that was rather too small for the job. Extraordinarily, we  managed to fit both families into this craft (4 adults, 6 children) and voyage down the Waveney from Geldeston Lock Inn. Although, we survived this experience, it was not to be repeated with such a full load.

Rodney and I also took the craft out from the Walberswick slipway. It was on an ebb tide, so going out was easy. It was an interesting, almost frightening experience coming back in. What we really hankered for was a fishing boat with inboard diesel engine. The boat below would have had our friend, the late Rodney Banks, drooling.

A real fishing boat

Near the ferry station, I had waited for an older couple (mid 70s may be, but who can tell) who were speed walking/jogging down the river path. To my surprise, I meet these two again as I walk away from the Bailey Bridge. How much further had they traveled  than I and what was their route? I had gone just over half a mile (0.92 km according to my mapping). An equivalent route through the village would have taken them over 1.6 miles. The map below shows the probable route, but it's only a guess. I have included a possible alternative painted in yellow.

A walker's route

Notwithstanding, my flirtation with agriculture as a student, I have next to zero knowledge of flowers (or plants of any sort). The following pictures were taken in well shaded tall shrubs, bushes and bracken next to the path. The first are of foxgloves, but I have no idea of the other two. Any thoughts?

Foxgloves?

It is just after 8 am, and I'm taken by surprise at the first sounds of motor cars on the road out from Walberswick.

At way mark 10 of the route map for this morning's walk, I query the existence of a bona fide footpath. It is not shown on the OS map for the area.

Is this a footpath or not?

However, there is an official footpath sign (I should have taken a photo) and the picture below shows the track that I followed. I wonder if the farmer disapproves.

Unofficial footpath

I return to the caravan just after 9 am, definitely ready for breakfast. But I have forgotten to bring the muesli. Irritated and tired, I trudge off to the Tuck Shop. There are sounds of doubles tennis being played behind a thick hedge; all women - too early for men? There are very few poppies this season, but here were some.

Poppies near the Tuck Shop

We normally reserve Tuesday for a fish meal, but I am feeling lazy and, in any case, thunder storms are forecast. We are both delighted that Bryony has provided more than enough chili for two meals. In the meantime, I attempt to finish off Robert Galbraith's "The Cuckoo's Calling". Under this pseudonym J K Rowling allows her imagination for devious plots together with very real and imperfect characters full reign.

Patrick tells me that between 1 and 2 am we were treated to sheets of lightning. He was fascinated but not very comfortable. Typically, I slept on until the need for an early morning piss.

Wednesday, 19th

The usual healthy breakfast: fried bread, fried eggs and fried sausages. Can I hear Bryony screaming from afar. "What about your bloody stroke?" I finish Robert Galbraith, so it's time to buy fish. It has been bright and warm since 8 am, but the weather is not forecast to last.

We take the Walberswick ferry across the Blyth River to the Southwold side. In this tiny fishing port, you are entirely dependent on the night's catch for the variety of fish available. The variety at the first stall was not inspiring. I was really looking fresh mackerel. The second stall is closer to the Harbour Inn and is often my preferred outlet. It has rock eel. The stall holder advises baking in foil for 18 - 20 minutes. 

On our return I spotted my ideal fantasy for a life at sea. Fortunately, finances prevent me from indulging this fantasy.

The fantasy of a life at sea.

At last I get my chance to drink at the Bell Inn. Two pints bitter shandy was not quite what I intended and I certainly paid for it later in the afternoon. Patrick is in talkative mood and we cover learning about statistics, activities post OU degree and communications with the DWP. This last was unusually constructive. They were certainly not hassling him over the period between the end of his current OU module and beginning of the next. In this is it was about 6 months.

Patrick in talkative mood.

The rain returns in the afternoon, so I grab my second novel. This is a first birthday present from Guthrie in, probably, two decades, but it is also the first time he has been at home over that period. "Dead Heat" is by Dick Francis and son Felix. Intellectual stuff hey!

The evening meal of baked rock eel is exceptionally good, but I wish I had rubbed the fillets with garlic as well as rape oil and lemon. The rain continues as we go to bed.

Thursday, 20th

The weather is bright but the ground and plant cover are very wet. I decide against a morning walk. Dick Francis awaits. Our hero is the usual almost clever bod who is being hassled by unseen baddies.The main interest for me was that the putative girl friend for the hero was a viola player and freelance head of section for the RPO. The artistic highlight of this romantic interlude is a performance of Elgar's Enigma Variations.

These Variations contain a couple of short solos for the lead viola. I note that this was not mentioned in the book, despite the fact that our hero went to see his beloved perform twice. Curiously, I had the privilege of taking these solos in Great Yarmouth Minster on 15th June.

Sudoku: Naked Multiples in Excel

This blog post is about the modelling of Sudoku naked multiples in Excel. It concerned with the user input / decisions and how the model uses this information to evaluate its validity in respect of the puzzle in its current state.

The concept of naked pairs, triples, quods etc and how they operate in Sudoku is best explained by experts. I suggest Hodoku or Kristanix. The web sites address the issues from slightly different perspectives but both are very clear.

Sudoku dimensions

Sudoku  has three dimension (row, column and sub-grid). Naked multiples operate in all three of the dimensions, but can only be evaluated one dimension at a time. In this model the first user choice is to decide on the dimension he wishes to investigate.

Select the orientation of the Naked Multiple analysis

This registration process calls a copy of an incomplete Excel template and creates a model for solving a discrete element of the Sudoku puzzle. The user must link the template to the original puzzle and the latest state of play for the puzzle. (It is not strictly necessary to link it to the original, but it makes the presentation easy in terms of identifying progress made.) The initial view of the working paper is shown below. It is identical in the cases of each of the three dimensions.

Joining the template to previous parts of the model

From here forwards,the spreadsheet presentation depends on the dimension
orientation of the template that you initially selected. The choices range from 1 to 9. 

The choice of Column or Row number is obvious.

While for row and columns it is obvious which row or column number to choose, it is less so for the sub-grid numbering, The sub-grid template includes a numbering layout.

Sub-grid numbering

Analysis by Column

This explanation has a starting point equivalent to column 6 in the Sudoku grid above. The picture below shows the same column picked out for detailed analysis. It highlights the Sudoku identities 3, 6 & 8, which, in this case, make up a Naked Triple. The player must mark these 3 Ids with a "Y" in the green input grid.

The user marks up the green input grid.

The remainder of this worksheet is given over to validation and analysis.

Validation

This evaluates whether the player selected a bona fide naked triple and, if so, considers whether it leads to a solution of one or more cells. In keeping with many non numeric analyses the logic can be long winded. The diagram below marks each of the analytical groups of cells or ranges with an identifying letter.

Validation and analysis by column - with identifying letters

Task Id	Description	Functionality
A	Inspection column	This is simply a repeat of the column selected at the top of the template, but without the formatting.
B	Current Id's	For each cell within the column the Identities currently available to it are shown. This includes solved cells.
C	Id's for inspection	This shows each cell which contains an Id that has been specified by the played / user as for inspection.
D	Identity count	For each cell, a count of the number of identities available to it. Cells with a count of 1 are solved. If the count is zero, there is an error.
E	Selected Id count	This is a count of the identities within the cell that are also flagged for inspection.
F	Id count & selected Id count equal	Phase 1 of the validation, evaluates whether the number of identities available to the cell and the number of selected identities are equal. Where this is true, the cell is set to "Y".
G	Validation complete	This covers two small processes. The first is a count of the Task Id F processes set to "Y". The second evaluates whether this count is the same as the number of Sudoku Identities selected for inspection. If this is true the validation is complete and the cell marked "Y". Failure at this point effectively stops further analysis.
H	Inspection column - less triples	This is simply a repeat if Task Id A, without any Sudoku Id associated with the validated triple. This means that the cells for the triple itself are left empty.
I	Identities after Task Id H	This is equivalent to Task Id B, it shows only those Sudoku identities that left. Where appropriate the Sudoku Id cells is marked "Y".
J	Identity count after Task I	This shows the count of Sudoku Id's in each cell after the removal of the naked multiple. Of course, this count will be 0, for the naked cells themselves.
K	Solved?	This analysis answers the question of whether the removal of the Ids associated with the naked triple generates a solution to the cell - in effect a clear result. The value 1 indicates that it did and 0 it did not.
L	Inspection column - post analysis	The naked triple cells are reinstated within the inspection column.
M	Stop cell	This takes the sum of Task Id K. If it is greater than zero, it stops any further analysis in this phase.

In theory, the player can undertake more than one naked multiple analysis within a single dimension consecutively. The Excel model permits this but only if there is no clear result from the first phase.

In this case, we did generate a clear result in the 8th row of the column under inspection. This is shown in the output Sudoku grid.

The processes involved with naked multiple analysis by Row and by Sub-Grid are identical, but their shapes on the Excel page look different to take account of the orientation within the Sudoku puzzle. A diagram of the Sub-grid page shows how different the Excel sheet looks.

Note that this analysis, although successful in showing surplus Identities did not deliver a final result.

Sudoku: Solutions without User Input

The creation of a detailed Sudoku grid usually brings a wider range of user methodologies for the player to employ, but some times solutions fall out of the process automatically.

In Sudoku: To automate ... or not to automate we generated a detailed grid of the puzzle in which all the the potential solutions for the unsolved cells were presented.The process also verified whether or not further solutions to puzzle had been generated as a result of the puzzle.

Result Validation - matched

In this case the number of result cells brought into the analysis matched the number processed out of the analysis. Sometimes the method we use for excluding potential values leaves only one Sudoku identity left. It therefore becomes as result.

Result Validation - not matched

In this latter case, we could not be sure that the result generated by the exclusion analysis had, itself, been accounted for in relation to its own related cells. Accordingly, we repeat the process described in Sudoku:A Game of Control and Error Management until we achieve a stable starting point.

There are times, especially when you have nearly solved a puzzle, where the re-verification process used repeatedly leads to a final solution automatically without any specific contribution from the Sudoku player. The picture below demonstrates one such case. It shows the penultimate phase before the presentation of a complete solution.

The penultimate phase before a complete result.

Sudoku: To automate ... or not to automate

In this article, I consider some of the options available to a spreadsheet designer for the automation of the interactions between the user (the Sudoku player) and the software he is using to record his moves.

The level of sophistication of Sudoku players varies hugely. I first discussed the process and the methodologies used to govern the interactions in a paper Spatial Modelling Techniques in Microsoft Excel. This paper considered the beginner level interactions with the puzzle.

The jump between the detail needed for beginners and that intermediate users is a major one. The design route we choose to take has significance both for the influence the player has both on the route he takes to solve the puzzle and the level to detail he has to contribute. The overall approach remains to demonstrate

• the step wise route that any individual used to solve the puzzle, and
• verify by analysis that those steps were valid.

as outlined in Sudoku: Options for layout and analysis in a Spreadsheet.

The decisions on how to automate and the level of sophistication of that automation process derive from experience - not always happy experience.

A Manual Approach

A curiosity about Sudoku puzzles is that essentially the same analytical process can nearly always be considered from two or more opposing perspectives. This is like viewing a mountain from its base or from its summit or from another nearby mountain. For example one can examine a row of cells and for each cell ask the question is there any single cell for which only one Identity is permissible because all others have been eliminated for that cell. Alternatively, you can consider the Identity group and ask the question is there any single Identity where only one cell within the row is permissible. Of course, it is necessary to ask the latter question in respect of each column and each sub-grid. It is this latter approach which generated my first detailed analysis of the Sudoku puzzle grid.

The spreadsheet layout is shown below. It demonstrates a case the player identified a solution for Id 2 in one cell; in this case Row 6 Column 9 (R6C9). However to complete the task, the user/player is required to consider and, if necessary, enter a value into each of the cells with a deep emerald green background. 

He is invited to review each Id (9) from each dimension perspective (3) and in respective each position in the Sudoku puzzle (81) - potentially there are 9 x 3 x 81 = 2,187 cells to complete.

Unfortunately, functional though this approach is, it does not deliver a verifiable step in the solution to the Sudoku puzzle. In the context of the spreadsheet model as compiled at that time, a further step was required. This is demonstrated below.

As the heading within the diagram implies, Sub-grid exclusion eliminates those cells where Id 2 already exists within the sub-grid. Whereas Columnar exclusion eliminates those single cells within the row where the Id 2 is already fixed for the column. The final row in this diagram states that only one position is permissible for Id 2 and, of course, that position is R6C9.

At this point, I confess, my patience gave way and I begin to wonder whether my readers will even have got this far.

Raw data and its potential uses

I was guilty of two failures. The review process described did not deliver a usable result for Sudoku player and also the player had to repeat essentially the same process for each of the three dimensions. 

The revised process shown below eliminates the vertical and sub-grid dimensions. The horizontal dimension is much easier to follow in the user's imagination. This also excludes all cells where the result is known. These cells are marked automatically as 'Fix'd'. In addition if the Id for a row is known, the whole row is excluded from the analysis. The user is invited to review each Id in turn and manually exclude all cells where the presence of the Id in either the vertical or sub-grid dimension excludes its availability.

Finally, this analysis delivers a usable result. Instead of the 9 x 9 grid shown as "Grid Current Status" in the picture above, we have a 9 x 9 x 9 grid as shown below.

It shows all the essential detail about the state of the puzzle. Curiously, it does not answer the question proposed at the beginning of this blog. If it had the four Id's circled in red would have been presented as the solutions to their respective cells and presented with a green background.

While this detailed grid is a useful result, it still involves a substantial amount of work. Since the only user was likely to be me I set about automating the process still further.

Full automation

The automatic process are described in the blog Sudoku: Options for layout and analysis in a Spreadsheet. When the process is applied to to this Sudoku puzzle we get an output in the form shown below.

Careful examination shows the two outputs to be almost identical. The only change that I can see is almost certain to be a user error perpetrated by me.

Viewing a puzzle in Excel

These Sudoku puzzles are worked in within an excel workbook. This particular example may be downloaded at Observer Sudoku 20190525.

Sudoku: Options for layout and analysis in a Spreadsheet

To state the bleedin' obvious, a spreadsheet is a a two dimensional array of addressed cells. When working with an environment that has 3 or more factors that require analysis its presentation in terms of layout and organisation are of crucial importance.

A Sudoku is unusual in that it presents 3 dimensional factors (row, columns and sub-grids) in a 2 dimensional surface. A further confusion is that Sudoku uses that values 1 - 9, but these are not numbers and you cannot employ them in simple arithmetic. A pervasive view is that spreadsheets are exclusively for arithmetic purposes (like monetary accounting). I have always railed against that view and here I consider that options for modelling the solution to a Sudoku puzzle in a spreadsheet environment.

The Sudoku model and its Purpose

For many, the only purpose of an automated (or semi-automated) approach is to find a solution quickly. But you can do that by instinct and guesswork far more quickly than building a spreadsheet model. I thought that my model should demonstrate

• the step wise route that any individual used to solve the puzzle, and
• verify by analysis that those steps were valid.

There has been a long running debate among spreadsheet users and constructors about how much of the sheet should be visible when in every day use. This discussion is concerned with the integrity of spreadsheets in a commercial or administrative environment. Sudoku is a game and everything should be visible. My one concession to security is that I protect the individual worksheets in the process to prevent accidental changes to model verification elements, but they can easily be unprotected. There are no passwords.

How much detail?

This really depends on the sophistication of the analytical methods that any individual intends to employ.  Since it does not take long for enthusiastic players to get moderately sophisticated, I chose to include sufficient detail within the model to handle that. 

The puzzle for solution is normally presented as a 9 x 9 grid. For example

Most experts on Sudoku present the detail of a puzzle in the format below. There various forms of the method of presentation but they all have the same fundamental format of a large 3 x 3 grid each containing a medium 3 x 3 (the basic Sudoku cells) and each cell contains a 3 x 3 grid of the potential Identities. For presentation purposes, solutions are presented as large Identities that overwrite the lowest level of potential Id's.

In theory, I could mimic this in Excel, but it was clear that the verification analysis would become very difficult and muddled. The choice of presenting the available Sudoku Id's for each cell in the seemed both convenient for presentation and for the analytical verification that would be required.

Even so a presentation that potentially contained all the detail of the puzzle, was likely to be difficult to understand at first glance (see below). A visual key to the Sudoku cells would be a huge advantage.

I would to make use of the presentational aspects of Excel that were inherent within a standard package. This suggested that an approach as shown below would be both functional and quickly understood.

The fist level of detail need to generate the presentation is discussed at A Game of Control and Error Management

Sudoku:A Game of Control and Error Management

The game of Sudoku and the methodologies for its solution have been  analysed by widely academics and other experts (for example The New Sudoku Players' Forum). The purpose of this article is to consider how we can demonstrate the relationship between the elements of the Sudoku once a particular Identity has gained control of an individual cell.

The modelling environment for this discussion is an Excel spreadsheet. An incidental consideration is the notion that spreadsheets have a wider modelling capacity than number crunching or simple database management.

Introduction - Names and Lists

The puzzled is built in a 9 x 9 grid of cells, but, from the perspective of a spreadsheet analyst, this is a very difficult structure to handle. We like simple lists. Also all Sudoku experts use a row/column nomenclature to describe their methodologies and solutions.

Conventional cell names for a Sudoku puzzle.

Our list is in the form of single column starting with the names of all the cells in row one and proceeding through each row. This gives a complete list of each of the 81 named cell.

Next we create a table, associated with the list of named controlling cells, of all the cells related to the controlling cells.

The cells which are logically associated with each 'master' cell. 

For example, if cell Id 42 contains the result 2 then it follows, by the rules of Sudoku, that a list of 20 further cells may NOT contain the result 2. These are:

• In row 4 - cells 41, 43, 44, 45, 46, 47, 48 and 49
• In column 2 - cells 12, 22, 32, 52, 62, 72. 82 and 92
• and in the 4th sub-grid - cells 41, 51, 43 and 53

This analysis is repeated for each of the controlling cells within the list of 81 named list. The final table comprises 81 rows and 20 columns (excluding the name column).

Now we know the relationships between controlling cells and others in the Sudoku table, but the analysis depends both on whether the controlling cell contains a result AND the Sudoku identity of the result. In effect each Sudoku identity must be analyzed and computed separately.

The Analytical Process

The process for evaluating whether an individual Sudoku Id is still available as a possible solution to a cell is long winded. Each cell containing a result has been evaluated for its impact on its related cells (see above). The evaluation is now reversed. Each cell that is not a result cell is evaluated for whether a result cell eliminates the availability of that specific Sudoku Id. Inevitably, each Sudoku Id must be evaluated separately. See the table below. Where the summary table (the right hand column) is greater than zero, Sudoku identity 3 is not available to that cell.

In practice we start with a table of results that looks like this one immediately below. The row at the top refers to the individual Sudoku identities.

And end with a table of deletions that looks like this.

This is then combined with the original Sudoku puzzle to look like this,

This columnar presentation then has to be converted back to the standard looking Sudoku puzzle and will look as shown below.

My personal convention is to show result cells with a green background, the values of the original puzzle in red script, those that are user calculated in black script and the unsolved cells with a white background.

Hard Sudoku

Saturday, 4 February 2017 and the Guardian reports how the EU leadership criticise the US president for his ‘lack of respect’. Earlier this week Jeremy Corbyn had placed a ‘3-line whip’ on Labour MPs to support the 2^nd reading of the European Union (Notification of Withdrawal) Bill. Quite a number defied the whip but not enough. Typically only Ken Clarke from the Tory side voted against.

It is sad that so many of the UK’s MPs have forgotten that they were voted into Parliament to act in accordance with the views they presented at the General Election 2015. Of course they are entitled to change their mind, but they must be convinced of the rationale for changing it. I wonder whether they are following what they believe to be the voter arithmetic rather than being persuaded by the argument. If that is the case, they have failed fundamentally in their duty as MPs.

But enough of all that. Hard Sudoku is what this blog is about. I tried the Guardian Sudoku puzzle (No. 3663) and recorded my progress. This record uses the methodology developed in MS Excel about 10 years ago. The methodology is quite weird and long winded but, in this context, it is useful for explaining how and why I got stuck. I describe below the argument I employed for getting unstuck.

I had got as far as here using conventional arguments and methods. Assuming no mistakes in recording, this would represent a valid solution thus far. 

The record shows that I had reviewed all the unsolved positions in the puzzle without making any further progress. This also covered the examination of individual Sudoku Nos. and their relative positions in each of the 3 dimensions of the puzzle. There were only four numbers left (1, 2, 6 and 8) and the reasons for not making progress bears some examination.

	Top 3 Rows	Middle 3 Rows	Bottom 3 Rows	Left 3 Columns	Middle 3 Columns	Right 3 Columns
Id 1	No positions fixed	All positions fixed	Only 1 position fixed	2 positions fixed?	All positions fixed	Only 1 position fixed
Id 2	No positions fixed	Only 1 position fixed	All positions fixed	2 positions fixed?	Only 1 position fixed	All positions fixed
Id 6	No positions fixed	Only 1 position fixed	Only 1 position fixed	Only 1 position fixed	Only 1 position fixed	No positions fixed
Id 8	Only 1 position fixed	No positions fixed	2 positions fixed?	2 positions fixed?	No positions fixed	Only 1 position fixed

Both Id’s 1 & 2 have a combination groups of three such that one group is solved completely and at least one other group where two positions are known, but not the third. Unfortunately, the information in the opposing dimension does not help us limit the position of the 3^rd ID. For ID 6, there is altogether too little information to make further progress.

ID 8 has 2 groups of three where the location of two of the elements is known. At first glance, the unsolved positions in the first column at rows 4 and 5 look the most promising. After all the sub-grid at the bottom left hand corner is solved completely and the sub-grid immediately above it has only these two elements unsolved. Nevertheless, no amount of examination has revealed a logically provable determination as to which cell should contain which ID.

I’m left with considering the position of ID 8 in the bottom 3 rows of the puzzle. At first glance this looks considerably less promising than the column 1 vacancies discussed above, especially since there are 3 unsolved ID positions for row 9 (see the values highlighted in green below) rather than the just two for column 1. However, in my solution methodology, I highlight cells where the possible solutions is limited to 1 of 2 IDs (see the cells circled in red).

The information available to us is therefore is

• ·         The location of ID 8 is known to be row 8 in for the bottom left sub-group and in row 7 for the bottom right sub-group, thus ID 8 must appear in one of the 3 cells in row 9 of the bottom middle sub-group. The right most cell of this sub-group row 9 has already been solved.
• ·         There are 3 unallocated cells in row 9 and 3 unallocated IDs.
• ·         But column 4 of the puzzle has two cells (named 2_43 and 8_48 in my nomenclature) whose ID values are limited 1 or 6. Logically, if these two values must appear in the two named cells, then they cannot appear anywhere else.

I conclude therefore the cell named 8_49 in row 9 must contain the ID value 8, since that is the ID value left for this column.

This is something of a tortuous argument but it seems to work. Any thoughts?