Str8ts

Str8ts is a logic-based number-placement puzzle distinct from but sharing some properties and rules with Sudoku. The solver is asked to fill the remaining white cells with numbers 1 to 9 (or 1 to n in puzzles with N cells per side) such that each row and column contains unique digits. Whereas Sudoku has the additional constraint of 3x3 boxes in Str8ts rows and columns are divided by black cells into compartments. Each compartment - vertically or horizontally must contain a straight - A Straight is a set of numbers with no gaps and in any order. For example,
[7][6][4][5] is valid but [1][3][8][7] is not.
Additional clues are set in the black cells - these numbers remove that digit as an option in the row and column. Such digits do not form part of any straight.

History

A hand made prototype of Str8ts which used black cells and the new rule of straights in compartments was invented by Canadian puzzle designer Jeff Widderich in 2007. He approached Andrew Stuart, a UK-based puzzle maker and programmer - to make the puzzle. Their collaboration settled how the clues would be determined and finalised the rules. The first puzzle was presented at the Nuremberg Toy Fair in February 2008. A daily puzzle has been published at [www.str8ts.com] since 24th of November 2008.

Setting the clues

Black cell clues are chosen in the following way: Given a completed str8ts puzzle there will be some unused digits in each row and column. This maybe zero, or it maybe several digits in some black cells. Where there is no remaining digit the black cell remains empty. From the available remainder digits a set is chosen that does not repeat a number.

White cell clues are determined in a similar way to Sudoku. From a completed puzzle numbers are continually deducted until the puzzle no longer contains a unique solution. The last unique solution should contain the minimum number of clues - although numerous different puzzles could be made depending on the order of the selection of cells. The black cell clues are taken into consideration when solving the puzzle in each removal loop.

Grading

As Str8ts belongs to the same class of puzzles as Sudoku the puzzle can demonstrate a wide spectrum of relative difficulty. The grade is determined by a combination of opportunities to solve at each stage and the difficulty of the strategy that grants each solution. An easier puzzle will have many places where a logical deduction can place a solution or eliminate a candidate number. When the whole puzzle is assessed in this way, plus some heuristics, a score can be determined. Over a large number of puzzles (>10,000) a bell curve of scores can be produced. This can be quartiled to group puzzles into specific grades..

Properties of Str8ts

The distribution of black cells - either symmetrically or asymmetrically - leads to a massive number of possible templates (Sudoku has one). Combined with the very large number of digit placements in the white cells leads to a very large number of possible puzzles. The density of clues in a good puzzle is similar to Sudoku and like Sudoku the number of clues does not determine the grade. A very difficult Str8ts puzzle might have many clues while one that unfolds easily might start with relatively few clues.

Strategies

The Str8ts puzzle unfolds through identifying the next cells that can be deduced from the clues in the black and white cells and the currently solved cells. On the easier puzzles this can be done by eye by scanning the rows, columns and the compartments. It is usually possible to find several opportunities to solve a cell since only one remaining number can fit. On the harder puzzles it is useful to make notes in some cells as a reminder of the options.

Solving sequence

To start solving Str8ts work in the smaller compartments ( 1, 2, 3 or 4 spaces) first. Using all the rules and clues and strategies listed below, the smaller compartments are usually easier to solve. Once solved, these smaller compartments start to fill in portions of the longer straights, and the filled in numbers remove themselves in rows and columns as possible numbers from the larger compartments. By adding additional numbers to the puzzle we essentially are turning larger straights ( 5, 6, 7, 8 and 9 spaces ) back into manageable compartments that will be easier to solve.

Compartment check

A Compartment of 5,6,7 and 8

A compartment is a set of white cells in a row or column bounded by the edge of the puzzle or a black cell. They will be between two and nine cells long. The rules state that all the numbers in a compartment must form a 'straight' - no gaps in other words - but the order will be unknown until you complete the straight. Given four cells with 5x8x, as in the example, we can know that to fill the gap we must use 6 and 7. It is possible to immediately eliminate 1,2,3,4 and 9 from those cells since they can never reach the clues we know about. Fortunately we have another clue in this example. The 6 tells us the rightmost X can't be a 6 so it must be a 7. That allows us to put the 6 between the 5 and 8.

When you are considering a straight it is necessary to check the rows and columns for any useful numbers. If they exist elsewhere then they cannot belong to any cell the number can 'see' in the compartment.

Black cell clues are interesting since they can cut the options in half. Say there was a black cell with '5' and it intersected a compartment such that it saw all the cells in the compartment. Let us also say the compartment is four cells long. Now, because 5 is in the middle this compartment of four it can only be 1/2/3/4 or 6/7/8/9. If there is a known you can decide which. So black cells, along with existing clues and solved cells will help pin down the remaining options.

Stranded digit

Stranded Digits

The Stranded Digit strategy comes from Compartment Check as it is worth looking for in it's own right. The Stranded Digit is easily understood - take the example on the right. Looking at the second row starting with the black clue 5 it cuts through a set of possible numbers for the two-cell on that row. Because the 5 is not possible (the clue) The 4's have become isolated from the 6,7,8 and 9s. That means we can discard 4 as a candidate. Similarly in the third row the 8 has isolated the 9 from the green cell. It leaves only the 7 and that is the solution.

Another way to look at it is this. Write out all the remaining numbers available in a compartment. If there is a gap - then only the numbers before or the numbers after the gap can be the solution to all the cells.

Lets take this small section and see how it plays out:

The 7 forces the 6 to the right. Above the 7 the 9 is too far away from 7 to be used - so 8 must fit there. 6 is disallowed because of the black cell clue above. And finally we can insert 7 into the last cell (marked in green).

Naked pairs

There is some cross over from Sudoku in this puzzle. Since rows and columns have the same rules as Sudoku they have some of the same strategies. The simplest is Naked Pairs. If two cells on the same row or column have the same two candidates - such as 1/6 and 1/6 - then we can be sure that 1 and 6 will appear in both those cells. We just can't be certain which way round yet. But that is extremely useful. If 1 and 6 are bound to be on those cells then any other 1 or 6 can be removed from that row or column.

The Naked Pair acts like a single cell clue - but make sure you apply it to only the row or column common to both cells. The two cells need not be next to each other or in the same compartment.

In the example we have three Naked Pairs - all aligned on different rows. The top row is the Pair 2/3. This attacks the cells at the start of the row (highlighted in red). Likewise the 7/8 on the second row effects the far right-hand cell. And finally the 2/4 on the third row gives us a solution - marked in green.

Naked triples

This is the same strategy as Naked Pairs except we are considering three cells, not two. For example, if 1/3/6 was present in three cells we know that 1,3 and 6 must be found in those cells – we just can’t know which way round at this point. So any other 1, 3 or 6 in the row or column can be discounted.

The trick with Naked Triples is that we don’t need all three numbers to appear in all three cells. Consider three cells with these numbers:
[1,3], [1,6] and [3,6]
Think of any combination from these pairs and you will always use up all three numbers in those cells. The rule really is – “If any three cells contain at most three numbers – then those numbers can be removed from the rest of the row or column”.

Publishing and commercial use

Str8ts, the name and the puzzle, is the property of Jeff Widderich and Andrew Stuart (Syndicated Puzzles). It is available for commercial reproduction in newspapers, magazines and web sites.

External links

• str8ts.com (official site) - two Daily puzzles and a monthly archive
• str8ts.de (official German site)
• str8ts.nl (dutch site)