Classics Collection

Classics Collection: Easy as ABC

Hi folks,
I am hoping to host another Beginners Contest soon, covering five more classic puzzle styles. That said, here is the next set of our puzzle series: Easy as ABC. This puzzle type goes back as far as I can remember, and it is as basic a logical puzzle style as I can think of.

Rules: Enter the given letters into the grid, so that each letter appears exactly once in each row and column. The letters outside the grid indicate the first letter in the respective row or column, seen from the respective direction.

Example and solution:

As usual, I will drop a couple of lines on solving techniques below, but before we get to that, a few general words about this puzzle type. The set of letters which must be entered is somewhat flexible, and some authors see fit to name the puzzle type accordingly (e.g. Easy as ABCD). For any kind of event containing more than one puzzle of this type, however, this might lead to confusion. Therefore I prefer to display the required letters somewhere next to the puzzle grid, as in the PDF file.

The difference between the grid size and the number of letters has an impact on the solving logic. Either too many or too few letters tend to make the puzzles boring. For instance, if there are as many letters as rows/columns in the grid, the structure is a straightforward Latin Square, and the clues outside the grid provide information only for the cells along the edges. This does not leave much room for solving techniques; in fact, such puzzles rarely have a unique solution.

On the other hand, if there are very few letters, there is hardly anything to enter at all. Personally I think that, starting with a grid size of 5×5, about two blank cells in each row and column are ideal. If the puzzles are very large, it may be a good idea to work with fewer letters, but then, I cannot imagine very exciting puzzles of this type of dimensions beyond 7×7 at all.

Next, regarding notation. Unless there is a great hurry, I find it advisable to not only enter the letters one has located, but also to mark the blanks. They are usually crucial for the solve, and there is no need to memorize all the positions inside the grid which cannot accomodate a letter for one or another reason. At least in the early stages, it is worth taking the time.

The point is, very often one cannot enter even a single letter right away. For example, with two blanks per row and column, each clue indicates that the respective letter must appear in the first, second or third position (with only blanks before it), but there is no telling which it is. A definitive result can only be obtained from clue intersections, and this usually requires locating a bunch of blanks first.

The simplest scenario occurs when there are two different clues next to the same corner cell, as in the bottom-left corner of the first puzzle of the set. (An entry of either A or C would violate the other clue.) There may be a successive application of this argument: The blank at R1C5 of the second puzzle implies another at R1C4.

Please note that the opposite direction does not work. Even if there are two matching clues pointing at a corner cell, it does not yield a certain entry. It may still be possible (and, assuming uniqueness, even necessary) to leave the corner cell empty to arrive at the solution of the puzzle.

Apart from the corner argument, there is another basic reason why a cell must be blank. Have a look at R4C5 in today’s first puzzle. This cell can only contain the letter A because of the horizontal clue. The cell is too far away from the vertical clue, though; it is impossible to place two more letters behind it.

The “long distance” argument can be used every time there are more cells between the secondary clue and the cell in question than blank cells are allowed. In fact, it seems like a good idea to compute – and somehow visualize – the exact number of blanks each row and column must contain as the very first solving step for every puzzle of this type.

The above technique can also be applied to other rows and columns at later stages. For example, look once more at Row 4 in the same grid. We already know that the corner cell R5C1 cannot accomodate an A, and the position R4C1 is equally impossible (in this case, the vertical clue is the “primary” one, and the horizontal clue is the long distance clue).

At this point, we have already found two blank cells in the leftmost column, hence the A can be entered in R3C1. In general, once the maximum number of blanks in a row or column has been located, every other cell must contain a letter. And if that row/column comes with a clue, the first entry is automatically known.

A variation of the “long distance” argument can be seen in Column 5 of the second puzzle. A blank at R2C5 would imply another at R3C5 because of the vertical clue. But that would be impossible; at least one of the first three cells must contain a letter. The letter C can thus be entered in cell R2C5.

It is noteworthy that clues outside the grid serve to eliminate certain letter positions even without the intersection part. (This is kind of similar to Skyscrapers.) Especially when several instances of a specific letter have already been entered, the “long distance” step can help determining the remaining cells which contain the same letter. This can be quite useful near the end of the solve.

There is more. Consider a row or column at the edge with many clues pointing into the grid from this side. A letter which is not among the clues has very few potential locations. If some more cells are eliminated from clues at the opposite edge, perhaps only one position is left. The C in the bottom row of the third grid can only be entered in R5C2 or R5C5, but the former is impossible due to the vertical clue in Column 2. Something similar applies to the location of the letter B in the topmost row.

This step occurs a great many times in this puzzle style (look out for edges with many clues, such as the bottom row in the fourth puzzle). In extreme cases, the long distance argument may not be necessary at all: The top row in the fifth grid leaves only one possible location for the letter B. At a later stage, this kind of step may work for other rows and columns as well.

There are a few other interesting constellations, but I will mention only one of them. Given several identical clues on the same side of the grid, they must somehow “share” the first locations between them. For example, the three B clues in the third puzzle from the South mean that one of them has the letter B in the first position, another in the second position and finally one in the third position.

This is a little like an X-Wing (or Starfish) in Sudokus. As a consequence, the three bottom rows cannot contain B’s in any other column than the ones mentioned before. Hence the remaining occurrences of the letter B must all lie in the block R1C2/R1C5/R2C2/R2C5. Again, this is very helpful if the inferences can be combined with clues from other edges.

Apart from what I wrote above, this puzzle type holds no major secrets. The puzzles 11 and 12 of today’s set are a little harder, but all the other puzzles can essentially be solved with these few techniques. Have fun!

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