Str8ts
Str8ts [streɪts] is a type of logic puzzle that has similarities with Sudoku . With Str8ts, too, a 9 × 9 grid is filled with the digits 1 to 9 in such a way that each digit appears only once in each column and row. In contrast to Sudoku, Str8ts also has black fields like in crossword puzzles . Only the white fields are filled. Black fields can be empty or filled in with a number.
The name "Str8ts" is derived from "straight", ie from the " street " in poker . “Str8ts” is pronounced like the English word “straights”. Contiguous white fields in rows or columns form a street with Str8ts, so they must contain a sequence of connected digits. The order is arbitrary.
A Str8ts puzzle consists of the 9 × 9 playing field with a pattern of black and white fields and some given numbers. Difficult str8ts can get by with very few given digits. In September 2010 there was a Str8ts with only two given digits.
The puzzle was invented in 2008 by the Canadian Jeff Widderich . He wanted to design a logic puzzle with similarly simple rules as Sudoku and a similarly complex logic. His idea was to insert the black fields and to replace the "block rule" of Sudoku with the "street rule" of Str8ts. The result is a puzzle that is comparable in complexity to Sudoku and also takes on the aesthetics of crossword puzzles.
For the implementation, Jeff Widderich teamed up with Andrew Stuart, a British programmer whose Sudoku website SudokuWiki.org is considered one of the best.
Where can you find Str8ts?
On-line
- On the website of the inventors there is a symmetrical and an asymmetrical new Str8ts for online play every day, a simple beginner Str8ts, mini-Str8ts in three levels of difficulty and an extremely difficult Str8ts every week.
- The German Str8ts website also offers the two daily Str8tsfc. In addition, there is a forum with explanations, solution strategies, special puzzles and discussions on extreme strikes.
- The WAZ online page has a daily update and a 30-day archive.
- The website and iPad app of the Neue Osnabrücker Zeitung (since November 2013, daily)
- The page of the Augsburger Allgemeine (daily)
Print media
- The Süddeutsche Zeitung has published a daily Str8ts on its puzzle page since March 2010. The SZ puzzles are very suitable for beginners, but there are rarely particularly difficult puzzles.
- In the Münchner Merkur and its regional editions a light Str8ts riddle has been published every day since December 2011.
- iPad app from Frankfurter Rundschau (since September 2010, daily)
- The daily newspaper Rheinpfalz has published a weekly Str8ts in its Saturday edition since August 2010.
- Zürcher daily newspapers Zürcher Unterländer and Zürichsee-Zeitung (since May 2011, weekly)
- Weekend supplement of the Nürnberger Nachrichten (since November 2010, every two weeks)
- Swiss television magazines TV2 and TVvier (since December 2010, bi-weekly and monthly)
- Freetime magazine exclusive (since April 2010, monthly)
- Lebanese newspaper The Daily Star (since January 2011, daily)
Since August 2009 STR8TS available as iPhone - application available for download since early 2013 as Android appliqué. There are now several books with Str8ts puzzles and a board game made of wood.
Rules and terms
The game consists of a grid with 9 × 9 fields, so a total of 81 fields in 9 rows and 9 columns. Some of these fields are black and some are white. Contiguous white fields in a row or column form streets .
The aim of the game is to complete the empty white fields of the puzzle. Black fields are not filled out. As long as the Str8ts is not solved, there can be several possibilities for different digits in a field. If these possibilities are noted, they are called candidates .
As for Sudoku, Str8ts does not require any math skills to solve it .
The following rules apply when filling out:
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Since there are also black fields that are empty, not all digits from 1 to 9 have to appear in every row or column. An empty black field can represent another missing digit in the row direction than in the column direction. The numbers 1 and 9 are not adjacent, so the sequence “9812” is not a valid street.
Difficulty levels
The previously published Str8ts have the following difficulty levels:
| Stars | German | English |
|---|---|---|
| * | Light | Gentle |
| ** | medium | Moderate |
| *** | Heavy | Tough |
| **** | Devilish | Diabolical |
| ***** | Extreme | Extremes |
For beginners, there are particularly simple str8ts called "easy" or "easy". In the case of light and many medium strikes, one can gradually see the solution numbers immediately. Complicated logical considerations are not required. In the case of the more difficult strikes, however, solution numbers are only revealed after the number of candidates in a field has been reduced to the solution by combining various logical considerations.
With strikes of the difficulty levels “devilish” and “extreme”, candidates often have to be determined systematically and gradually reduced. The more complex solution strategies must also be used to solve a puzzle.
Some of the “extreme” strikes can not be solved with the deductive solution strategies described so far , although they are valid, ie have a clear solution.
The determination of the difficulty of a strike is neither clear nor undisputed. The perceived difficulty depends to a large extent on whether or not you can quickly recognize a logical connection. The assignment to levels of difficulty can be determined by a solution program by counting which solution methods have to be used and how often until the complete solution is determined. Weighting factors are assigned to the various solution methods . The result is a total score, which then determines the level of difficulty.
variants
The normal Str8ts is a square of 9 × 9 fields, some of which are black. The black fields can be arranged as desired. In typical str8ts, about 20 fields are black, the range is from 1 to 35.
- Symmetrical Str8ts . Here the pattern of the black fields is point-symmetrical around the center of the playing field. Such strikes look harmonious, but they are no different from asymmetrical strikes when solved.
- Asymmetrical str8ts have black fields with an irregular pattern. Str8ts with a particularly high level of difficulty are more often asymmetrical.
- Mini-Str8ts consist of a 4 × 4 or 6 × 6 field. The only rule difference: only the digits 1 to 4 or 6 are used. Mini-Str8ts can be solved quickly and usually without writing down candidates.
Transformations
The same puzzle can be represented in different ways through different transformations. There are three transformations of the pattern and one digit transformation. The individual transformations are:
- Rotation by 90 °
- Rotation by 180 °
- reflection
- Mirroring the sequence of digits. This means that the number N is replaced by the number 10-N (i.e. 1 by 9, 2 by 8, etc.). Any exchange of digits, as in Sudoku, is not allowed because of the street rule. The digits cannot be replaced by symbols or colors either, because then the property of the ordered sequence would be lost.
The four transformations can be combined as desired and applied to a certain Str8ts, the result is always a correct, apparently new Str8ts puzzle. In fact, it is still the same Str8ts in a new representation. 16 variants can be generated through transformations.
Solutions
A systematic approach and logical thinking are required to solve problems . This is the only way to get a complete solution step by step. Light struggles can be solved in the head through logical thinking. For more demanding puzzles you can no longer do without notes in order to write down different possible solutions for each field, the candidates.
First you search the Str8ts for streets that only contain an empty white field. This can be a two-way street with a given number or a longer street in which only one field is empty. According to the 3rd rule, the street rule, only the two neighboring digits of the specification come into question in a two-way street. If the default number is 1 or 9, there is even only one neighboring number, which is then entered as the solution of the free field. If two possible digits have been found, one of them can often be excluded because it violates the second rule, the row-column rule. Very simple strikes can be solved completely in this way.
The basic principle of all possible solutions becomes clear: The starting point for solving every empty white field is the list of candidates, which initially always contains all digits from 1 to 9. Using the solution strategies, the candidates in a field are gradually deleted. If only one candidate remains, you have found the number of the answer for the field. If there is no candidate left, you have made a mistake or the puzzle is faulty.
If you can't find another number, you first have to identify the candidates for the fields and then gradually reduce them. Which candidates are possible for a field can be noted using the methods known from Sudoku. It is advisable to start with fields for which apparently only a few candidates exist and to write them down. You can see further logical connections so that candidates can be deleted. For example, there are only a few candidates for a two-way street with a given number, but even with three-way streets, the number of candidates is small due to the direct application of the first three rules.
In the case of very difficult strikes, it can happen that the candidates for all empty fields have to be determined and noted before a solution number is discovered after applying various solution strategies and the resulting reduction in the number of candidates. If a new digit is found, its effect on the candidates for the fields in the same column or row can be entered. If the candidate list has decreased in a field, it follows from the 3rd rule that further candidates can possibly be excluded in the fields of the associated streets.
Immediate application of the rules
Row-column check is the immediate application of the 2nd rule. For a field, you can exclude all digits that already appear in the row or column of the field. This method is obvious and is therefore used intuitively without being perceived as a method.
Road testing means first of all the immediate application of the 3rd rule. For example, if you find a street of three with the numbers 3 and 5 and a free space, then there must be a 4. The road rule also leads to other important conclusions, so it is at the heart of the entire game.
Because of the black fields, not all digits from one to nine necessarily appear in a row or column, as is the case with Sudoku. In the course of a solution there are therefore safe or possible digits among the candidate digits . For some of the solution strategies, it is of crucial importance whether a digit is certain or whether it is only possible .
- Possible are all candidates digits that have not yet been ruled out.
- Sure are candidates digits, which are known to occur must . Here it becomes even more complicated, because a candidate digit can certainly be in its row street, but it can only be possible in its column street . There can also be candidate digits that are not certain in their streets, but must be certain in the line. Therefore, depending on the solution method, a precise distinction must be made as to whether a digit is safe in its row street, column street, row or column.
Basic solution methods
Road test
In addition to the immediate application of the 3rd rule, the street check (compartment check) leads to the exclusion of candidates, both inside and outside the respective street.
From the known digits or the candidates in the white fields of a street, it is determined within which limits the digits of the street can lie. All digits outside of this area can be crossed out of the street fields.
If the value range of the street is less than double its length, then there are digits that must definitely appear in this street. Safe digits can be deleted from the fields outside the street in the row or column of the street. If, for example, the number 4 is in a field in a street of three, then only the numbers 2, 3, 5 and 6 can appear in the two empty fields from the street rule alone. All other digits can be crossed out in the fields of this street. The digits 2356 in this example are then called possible digits.
If the number 2 is omitted in both unresolved fields in this street, the 3, 5 and 6 remain in addition to the already known 4 as possible numbers of the street. That means the street contains either 345 or 456. In both combinations there is also a 5 in addition to the 4. The 5 is then safe called number (Necessary digit) of the road. Recognizing secure digits is critical to some solution methods. A safe digit of a street can be deleted from the candidates of all external fields of the row or column to which the street belongs. Since the example street is part of a line, the 5 can be deleted from all fields in the line outside the street.
If a road does not yet contain a known solution, the road test can be carried out in the same way on the basis of the currently possible candidates. For example, if the street of three contains the candidates 3456-35-456, then 3456 are the possible digits and 45 are the safe digits of the street. 45 can be deleted in their line outside the street.
Another example: the street of three with the candidates 23456-56-46 has the possible digits 3456, 45 are secure digits. The 2 is not possible because it cannot form a street of three with the digits 56 in the second field.
The numbers 1 to 7 appear in the five-way street shown. So 3, 4 and 5 are safe digits, they have to appear. The 1, 2, 6 and 7 can, but do not have to occur.
Andrew Stuart describes the road test in two sections in the “Strategy discussion” section of his website. He calls the first part, which delimits the digits within the street, “Compartment Check”. The second strategy, which excludes safe digits outside of the street, he calls "high / low".
Hidden single digit
A hidden single digit is a safe digit that only appears in one field in your street (the 3 in the example opposite). In this field all other candidates can be deleted, the hidden single digit is the solution of the field.
It must be a secure number. If the single occurring digit is possible but not certain, then it is not a safe solution either.
Stranded digit
If a digit with the possible digits of the other fields of a street does not form a valid street, then it is stranded and can be deleted (stranded digit).
Example in the picture on the right: Dreier-Strasse candidates are 2457-456-4567. The 2 is stranded because there is no 3 required to connect to the digits of the other two fields.
Even / Odd pairs
If a field of a street of length 2 only contains even digits, you can cross out all even digits in the adjacent field. The adjacent field can only contain digits that differ by 1. The same applies to the odd digits.
Long roads
Streets with a length of 5 to 7 fields inevitably contain certain digits and thus exclude them in other streets in the same row or column:
| length | Exclusions |
|---|---|
| 5 | 5 |
| 6th | 4, 5, 6 |
| 7th | 3, 4, 5, 6, 7 |
If there is also a digit in a black field in the row or column, you can exclude further digits.
Other solution methods
The basic solving methods described above can solve most simple to moderate problems. The following solution methods are used less often, but are necessary for severe, diabolical or even extreme strikes.
Large gap rule
If there is a pair of digits in a field of a street, the difference of which is at least as great as the length of this street, then these two digits can be crossed out in all other fields of this street. This is due to the fact that neither can exist together in the street, nor can they appear twice. Each of the two possibilities therefore excludes the occurrence of both digits at any other point on the street. Neither of the two digits can therefore be "certain".
For example, assume a field with the pair 19 in a street is shorter than nine fields. If there were a 1 anywhere else in the street, the 9 should be there. If there were a 9 somewhere else, there would have to be a 1, but neither is possible, since a street with eight or fewer spaces cannot contain 1 and 9 at the same time.
Rule of the number of rows / columns ("Setti rule")
From the Str8ts rules it follows that each digit must appear as often in columns as in rows. So if a digit does not appear in any line, there must also be a column in which it does not appear (and vice versa). If you z. If, for example, you know that a digit appears in exactly seven lines, then you know that it must also appear in exactly seven columns. If this number is determined in seven columns as possible and in two columns as possible, it can be deleted in the two possible columns. If it is determined to be certain in six columns and as possible in one more column, then it is certain there, i.e. This means that other candidate numbers are omitted there.
To apply the Setti rule, the entire playing field has to be examined, which makes this solution method quite time-consuming. For really extreme struggles, however, the Setti rule is often a very effective solution strategy. When used, the candidate digit property "safely in the row" or "safely in the column" must be considered.
Naked groups
Pair (naked pair): If only two identical candidate digits appear in two fields of a row or column, then these two digits can be deleted from the remaining fields in the row or column.
The pair method as well as the triple and quadruple methods described below work in rows and columns of Str8ts in the same way as in rows and columns of Sudokus. Pairs, triples or quadruples can occur in different streets and can be formed from all candidates (possible or certain).
Triple (naked triple): If only three identical candidate digits appear in three fields of a row or column, then these three digits can be deleted from the remaining fields in the row or column.
The triple can contain fields with the three or two of the three digits. Example: any three fields in a column contain 12-23-123. Then there is a triple 123.
Quadruple (naked quadruple): If only four identical candidate digits occur in four fields of a row or column, then these four digits can be deleted from the remaining fields of the row or column.
The quadruple can contain fields with the four or two or three of the four digits.
Hidden groups
A hidden pair exists when two secure digits of a street only appear in the same two fields of the street. The other candidates in these two fields can be deleted.
Example: In a street of five with candidates 124-23567-467-12356-47, the digits are 345 secure digits. Since 3 and 5 only appear in fields 2 and 4, the remaining candidates can be deleted. Result: 124-35-467-35-47.
A hidden triple occurs when three safe digits of a street only appear in the same three fields of the street. The other candidates in these three fields can be deleted.
A hidden quadruple exists when four safe digits of a street only appear in the same four fields of the street. The other candidates in these four fields can be deleted.
X-wing group
X-Wing: If a certain number appears exactly twice in two columns, namely in the same two lines, and if the number is a number that is safe for the respective column street, then it can appear in both lines excluded elsewhere.
The rule applies in the same way to safe digits in row streets. It is important that a digit in a row can be safe and at the same time possible in the column , but not safe.
The four fields in which the safe digit occurs form a rectangle in which the digit must appear as a solution in one of the two diagonally opposite corner pairs. The two diagonals form the X, which, as with Sudoku, has led to the name X- Wing as a solution method.
If an X-Wing of safe digits z of the column exists, digits z that were not previously safe but belong to the same row of rows become safe digits. The possible range of values for the line street is reduced. This also applies to X-Wings in the line direction.
An extension of the X-Wing logic to three rows and columns is called Swordfish (English for " swordfish "): If a certain number occurs two or three times in three columns, in the same three rows, and it deals with it If the number is a safe number for the respective column street, then it can be excluded elsewhere in the three lines.
If a Swordfish of safe digits z of the column exists, digits z that were not safe before, but belong to the same row of rows, become safe digits. The possible range of values for the line street is reduced.
Here, too, the rule applies in the same way to safe digits in street streets.
The extension of the X-Wing and Swordfish logic to four rows and columns is called Jellyfish (English for " jellyfish "): If a certain number occurs two, three or four times in four columns, namely in the four same lines, and if the digit is a safe digit for the respective column street, then it can be excluded elsewhere in the four lines.
If a jellyfish of safe digits z of the column exists, digits z that were not safe before, but belong to the same row of rows, become safe digits. The possible range of values for the line street is reduced.
Here, too, the rule applies in the same way to safe digits in street streets. The principle can also be extended to 5 × 5 ("Starfish", English for " Starfish "), etc., as long as it is only applied to safe digits.
Hypothesis and contradiction
If you can't go any further, a hypothesis (what-if ?, trying out, Ariadne's thread , trial and error , backtracking ) helps , which is followed up until it leads to the solution or a contradiction . It should only be used if all of the above methods no longer help.
Fields that only have two candidates are particularly suitable for trying out, because a wrong hypothesis then confirms the alternative as correct. You have to remember the starting point of the assumption. If the pursuit of the assumption made does not lead to a contradiction, one pursues the assumption of the alternative; if this leads to a contradiction, the first assumption was correct. As a special situation it can arise that all assumptions in another field result in the same number as a conclusion. Then you have found a number at this point.
With hypothesis and contradiction, every problem, no matter how difficult, can ultimately be resolved and its uniqueness can be demonstrated. The way there can, however, be very tedious and confusing if one has to assume further sub-hypotheses when pursuing a hypothesis. Some players generally reject hypothesis testing as illogical or unaesthetic.
Logically solvable?
The statement that a Str8ts can be solved logically leads again and again to discussions about the question of whether hypothesis and contradiction are viewed as a logical method.
The methods described above can all be called deductive . A method is considered deductive if a conclusion can be derived from one or more premises.
The row rule, for example, is deductive because it can be formulated like this:
IF digit n occurs in the line, THEN digit n cannot be a candidate in an unresolved field of the line.
Or more generally:
IF pattern occurs, THEN certain digits can be excluded.
In this sense, the hypothesis-contradiction method does not follow any deductive logic. With the statement “The Str8ts can be solved logically” it is meant that the Str8ts can be solved with deductive methods.
Programmable solutions
A solver can use the deductive methods described to solve a problem. The methods are gradually applied to the entire Str8ts. Whenever a solution is found or candidates can be excluded, a new program loop begins. In order to really solve all the strikes, backtracking must also be used as the last method .
A solver can be found on the websites www.str8ts.com and www.str8ts.de. However, it does not contain all of the methods described above, especially not the final backtracking. Therefore he cannot solve all problems.
Backtracking method
On the computer you can also solve a Str8ts exclusively with the backtracking method. Starting with the first free field, all candidates are systematically tried. At the first contradiction you go back ( backtrack ) and choose the next candidate. If a field has no more eligible candidates, the system goes back one field and the next candidate is chosen there. This approach can be formulated very elegantly, recursively , and you can be sure that all possible combinations are searched. Since there can be thousands of paths, this algorithm is only suitable for computer programs .
Help with solving
The tried and tested methods for remembering candidates in Sudoku can also be used in Str8ts.
The "candidate list"
Starting with the short streets, you determine the candidates field by field and write them as small numbers at the top or bottom of the field. If a candidate is excluded, they are crossed out.
If a digit in the candidate list becomes a safe digit, it can be underlined if it is safe in the row street or a vertical line next to it if it is safe in the column street. The safe candidates in both directions can also be marked with colored dots in two colors.
This method can get confusing with small print and difficult strings. Then only copying and enlarging the puzzle will help.
The "clock-hand method"
For small print strings in newspapers, the clock-hand line method is useful to keep track of candidates for a field. Make a small line in the field at the place of the "clock hand" (see picture). The five is an exception; it is shown as a small point in the middle. This way you can remember several candidates for one field. If you don't have an eraser to hand, cross out a candidate line if further considerations rule it out. This method is considered more legible than writing small digits. The marking of secure digits can again be done with two colors.
Write down points for candidates
You can set small dots like a telephone keypad and thus note possible candidates for a field, starting for the one in the top left corner. At the top in the middle is the point for a two, in the upper right corner the point for a three, on the left in the middle is the point for a four and so on up to the point for a nine, which is then in the lower right corner Corner stands.
Create new str8ts
More difficult than solving a problem is designing one. Without the help of a solution program, it would be extremely time-consuming to actually try.
If you have a solution program, you can create a new Str8ts in the following way:
Create patterns
In the first step, a pattern of the black fields is created. The only real condition is that a Str8ts must have at least one black field.
Typical str8ts have 15 to 25 black squares. Further restrictions are often made. This includes the symmetry of the pattern, especially because the frequently used point symmetry around the center point leads to an aesthetically more satisfying pattern. A condition can also be that no streets of length 1 are allowed. Interesting streets have many long streets, difficult ones often have 3 or 4 streets in a row or column. If you adhere to such conditions when designing a pattern, specific patterns for easier or more difficult strings can be created.
Fill in the white fields
In the second step, the empty pattern in the white fields is filled with numbers in such a way that all rules are adhered to. This step corresponds to the solution of a finished string, but with the restriction that the empty pattern does not have a clear solution, but usually very many. But it is also possible that there is no solution at all for a selected pattern. The second step is therefore to find some solution for all white fields.
A solution program is used to fill in the empty white fields, which either uses the deductive methods described above or carries out backtracking.
Fill black fields
In the third step, the black fields are filled with digits as far as possible, whereby the row-column rule must be observed. Therefore, black fields can also remain empty here if there is no permissible number.
Empty white fields
In the fourth step, digits are gradually deleted from white fields. After each deletion, a check is made to determine whether the resulting problem can be resolved. This is repeated as long as the problem remains solvable.
If a program that uses the deductive methods is used for this check, then by activating or deactivating the individual methods, one can control which of the methods are required to solve the emerging new problem.
Empty black fields
In the fifth step, the digits are also gradually deleted from the black fields. After each deletion it is checked again whether the Str8ts can clearly be resolved.
Up to the fifth step, a program that uses backtracking can also be used to check the clear solvability.
Determine the level of difficulty
In the last step, the new Str8ts is solved with the deductive solution program, counting which method is used and how often to determine the solution numbers. The degree of difficulty is then derived from the total value determined in this way. If the solvability with backtracking was previously checked, it can now turn out that the deductive methods are not sufficient to come to a solution. In this case an extreme str8ts of the highest level of difficulty has arisen.
If it turns out that the level of difficulty is higher than desired, additional digits can be used as specifications. This then reduces the difficulty to the desired level.
literature
- Jeff Widderich, Andrew Stuart: STR8TS - The new addictive number puzzle , jezza! Verlag, Geltendorf 2009, ISBN 978-3-941969-00-1
- Jeff Widderich & Andrew Stuart: STR8TS , Süddeutsche Zeitung GmbH, Munich 2010, ISBN 978-3-86615-810-8
Web links
- Mathematical handicrafts : Jürgen Köller - An experience report about getting to know Str8ts
- German Str8ts-Forum : Forum on rule questions, strategies, special puzzles
Individual evidence
- ↑ English language website of the inventors of Str8ts
- ↑ German-language website of the jezza! Publishing house
- ↑ WAZ-Online on the game site ( Flash advertising with audio)
- ^ Str8ts-Player on the website of the Neue Osnabrücker Zeitung
- ^ Str8ts website of the Augsburger Allgemeine
- ↑ X-Wing in Sudokuwiki
- ↑ Sword-Fish in Sudokuwiki
- ↑ Jelly-Fish in Sudokuwiki