# Reading List: ‘Absolutely Nasty Sudoku, Level 2’ (Frank Longo)

I enjoy logic puzzles (not just in a formal context, to be clear). Probably one of the most well-known genres of these is sudoku, a grid-based constraint satisfaction problem made popular by the Japanese publisher Nikoli. Given a 9 by 9 grid which may be decomposed into three 3 by 3 sub-grids (boxes), fill in numbers such that there is precisely one number from 1 to 9 in each row, column and 3 by 3 sub-grid. Some numbers are already given as clues; there is usually only one unique solution for a given clue set.

Solving a sudoku puzzle, in general terms, is NP-complete (of course, for a 9 by 9 grid, solving can be done in constant time). ‘Obvious’ strategies involve finding the only place in each row/column/box where a given number could go (since each of these must contain each digit once), or finding the only number that could be assigned to a given cell.

The next step would then involve considering interactions between cells – for example, ‘these two cells in this row can only contain 3 and 5, so no other cells can contain a 3 or a 5’, or ‘both of the cells where 7 could go in this box are in the top row, so there can’t be a 7 in the top row of the other two boxes that are horizontally aligned’ (called a pointing pair). Complexity only develops from there; there are a myriad of clever strategies as shown on the sidebar here.

Sudoku (sudokus?) are often featured in newspapers, and there are many compilations of these. However, as one gets familiar with the strategies, these can become less interesting. For example, consider this hard sudoku from the Guardian; a not-that-smooth solve yielded a 5m11s time, and the most complex technique I used was the ‘pairs’ reasoning I outlined above.

I recently discovered Frank Longo’s Absolutely Nasty Sudoku series, and thankfully this is one series which seems to live up to its name. Consider this grid (which is the puzzle above, converted into an easier to read format):

The grid pictured above is from a puzzle towards the tail end of the Level 2 book. Although no cells are obvious (and it doesn’t look like the pairs/triples logic yields anything), there are several ways to proceed from here:

• In row 3, you can have an 8 only in columns 2 and 9. This is also true in row 6. This means that no other rows can have 8s in columns 2 and 9; in particular, the second cell in the bottom-most row can’t be an 8, and has to be a 4. (This strategy is called X-Wing.) This is arguably the most straightforward deduction here, though it wasn’t the one I saw.
• Consider that if row 5 column 1 was a 5, then row 8 column 1 must be an 8; that forces row 9 column 2 to be a 4, and row 9 column 7 to be an 8. Alternatively, row 5 column 1 can be an 8. Either way, row 5 column 7 can’t be an 8, and must be a 4. (This strategy is called an XY-Chain.)
• Suppose row 6 column 7 was a 9. Then, the rectangle of cells with vertices in rows 1 and 3 and columns 4 and 7 admits two solutions; without affecting other cells, I can have either a 3 or 7 be in the top left corner. Thus, row 6 column 7 can’t be a 9, and must be a 3. (This strategy is called a Unique Rectangle.) This was actually the next step I made when solving this puzzle!

A key criterion when evaluating whether a Sudoku book (or a puzzle book in general) would be the difficulty and quality of the puzzles. The difficulty level for Level 2 is well-placed for someone like me, if maybe a little on the easier side; I’m familiar with most/all of the ‘basic’ techniques, and am aware of though not confident with trickier techniques like the aforementioned Unique Rectangle. I might have preferred Level 3, actually; there were several puzzles at the beginning of Level 2 that seemed like they could be solved with techniques I’d consider as basic. I also haven’t found any “broken” puzzles which couldn’t be solved (presumably, the author would have checked these with a solver before publication)!

I also have other criteria for evaluating Sudoku books, though. I tend to write down cell candidates as well as other annotations when solving. These help me more easily remember observations like pointing pairs; I’d prefer the grids to be sufficiently large that I can comfortably write these out. The grids in this book were certainly of adequate size. While my handwriting tends to be quite small, I don’t think grid size would be problematic for most people.

The choice of ring binding seems to make sense, too; the book needs to remain open when one is solving. Generally I found the paper to be very smooth to write on. I don’t have many complaints here.

The book has a very brief foreword, and then gets straight into the puzzles (with solutions at the back). The target audience is clearly people who are well-accustomed to Sudoku. As mentioned earlier, this is the second in a five-part series and based on online reviews I determined this to be a good place to start. I wouldn’t recommend this for someone just starting out with Sudoku, though.