Cage combinations - Revision history

Ruud at 08:24, 30 October 2021

2021-10-30T08:24:07Z

← Older revision		Revision as of 08:24, 30 October 2021
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	A [[Killer]] ~~Sudoku~~ contains [[cage]]s and cage sums. Knowing the number of [[cell]]s in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the possible combinations of [[digit]]s that can go into the cage. For example, if we have a cage of 2 cells whose sum is 5, written 15[2], then either it contains the digits 1 and 4, or the digits 2 and 3. The cage combinations are written in set notation as {1,4} or {2,3}, or simply {14} or {23}.		A [[Killer Sudoku]] contains [[cage]]s and cage sums. Knowing the number of [[cell]]s in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the possible combinations of [[digit]]s that can go into the cage. For example, if we have a cage of 2 cells whose sum is 5, written 15[2], then either it contains the digits 1 and 4, or the digits 2 and 3. The cage combinations are written in set notation as {1,4} or {2,3}, or simply {14} or {23}.

	Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs\|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[eliminate\|eliminations]], either simply because the candidate is not part of any combination, or in conjunction with some other [[solving technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.		Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs\|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[eliminate\|eliminations]], either simply because the candidate is not part of any combination, or in conjunction with some other [[solving technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.

127.0.0.1 at 14:57, 24 October 2021

2021-10-24T14:57:38Z

← Older revision		Revision as of 14:57, 24 October 2021
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	Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs\|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[eliminate\|eliminations]], either simply because the candidate is not part of any combination, or in conjunction with some other [[solving technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.		Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs\|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[eliminate\|eliminations]], either simply because the candidate is not part of any combination, or in conjunction with some other [[solving technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.

	== Example 1 ~~<English version>~~==		== Example 1 ==
	For a cage 24[3], its only combination is {7,8,9}. Then we can eliminate the digits 1 to 6 from all three cells in the cage. Further, if all the three cells all belong to the same [[line]] or [[nonet]], we can then apply [[Naked Triple]] to that line or nonet.		For a cage 24[3], its only combination is {7,8,9}. Then we can eliminate the digits 1 to 6 from all three cells in the cage. Further, if all the three cells all belong to the same [[line]] or [[box\|nonet]], we can then apply [[Naked Triple]] to that line or nonet.

	~~== Пример 1 <Russian version>==~~		== Example 2 ==
	Для клетки 24[3], только комбинация - {7,8,9}. Затем мы можем устранить цифрам 1 на 6 из всех трех ячеек в клетке. Продвигать, если все три ячейки все принадлежат тому же [[line]] or [[nonet]], мы можем тогда приложите [[Обнаженный Тройной]] на эту линию или nonet.

	== Example 2 ~~<English version>~~==
	Given a cage 28[4], there are two possible cage combinations: {4,7,8,9} or {5,6,8,9}. Either way, the cage must contain the digits 8 and 9 in 2 of the 4 cells. Knowing this, we might be able to apply the [[Locked Candidates]] technique. Further, if a cell whose candidates are 8 and 9 sees all 4 cells, we may be able to apply the [[Killer Subset]] technique.		Given a cage 28[4], there are two possible cage combinations: {4,7,8,9} or {5,6,8,9}. Either way, the cage must contain the digits 8 and 9 in 2 of the 4 cells. Knowing this, we might be able to apply the [[Locked Candidates]] technique. Further, if a cell whose candidates are 8 and 9 sees all 4 cells, we may be able to apply the [[Killer Subset]] technique.

	~~== Пример 2 <Russian version>==~~		== Example 3 ==
	Данна клетка 28[4], есть две возможных комбинации клетки: {4,7,8,9} или {5,6,8,9}. Любой путь, клетка должна содержать цифрам 8 и 9 в 2 4 ячеек. Зная это, мы вероятно способными приложить [[Locked Candidates]](Запертые Кандидаты) техника. Продвигать, если ячейка, чьи кандидаты - 8 и 9 видит все 4 ячейки, мы возможно способными приложить [[Killer Subset]](Подмножество Убийц) техника.

	== Example 3 ~~<English version>~~ ==
	Suppose we have a cage 15[3] such that		Suppose we have a cage 15[3] such that
	* Cell A has candidates 3, 4, 5.		* Cell A has candidates 3, 4, 5.
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	Alternatively, we can also consider the minimum and maximum possible values of the sum A + B. Since 3 <= A <= 5 and 7 <= B <= 8, we have 10 <= A + B <= 13. Due to the constraint A + B + C = 15, we have 2 <= C <= 5. This means that we can eliminate 1 and 6 from cell C.		Alternatively, we can also consider the minimum and maximum possible values of the sum A + B. Since 3 <= A <= 5 and 7 <= B <= 8, we have 10 <= A + B <= 13. Due to the constraint A + B + C = 15, we have 2 <= C <= 5. This means that we can eliminate 1 and 6 from cell C.

	~~== Пример 3 <Russian version> ==~~
	~~Полагаем, что у нас есть клетка 15[3] так что~~
	* Ячейка имеет кандидатов 3, 4, 5.
	* Ячейка B имеет кандидатов 7, 8.
	* Ячейка C имеет кандидатов 1, 2, 3, 4, 5, 6.

	Мы могли бы перечислить перестановки (3+7+5 или 3+8+4 или 4+8+3 или 5+7+3 или 5+8+2) и решать, что 1 и 6 могут быть устранены из ячейки C.

	Кроме того, мы можем также рассматривать минимальные и максимальные возможные величины суммы + B. С 3 <= <= 5 и 7 <= B <= 8, у нас есть 10 <= + B <= 13. Из-за ограничения + B + C = 15, у нас есть 2 <= C <= 5. Это означает, что мы можем устранить 1 и 6 из ячейки C.

	~~Author: Email: Javof@mail.ru~~
	~~ICQ:462675542~~

	[[Category:Killer Sudoku]]		[[Category:Killer Sudoku]]
	[[Category:Solving Techniques]]		[[Category:Solving Techniques]]

127.0.0.1 at 14:55, 24 October 2021

2021-10-24T14:55:03Z

← Older revision		Revision as of 14:55, 24 October 2021
Line 1:		Line 1:
	A [[Killer]] Sudoku contains [[cage]]s and cage sums. Knowing the number of [[cell]]s in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the possible combinations of [[digit]]s that can go into the cage. For example, if we have a cage of 2 cells whose sum is 5, written 15[2], then either it contains the digits 1 and 4, or the digits 2 and 3. The cage combinations are written in set notation as {1,4} or {2,3}, or simply {14} or {23}.		A [[Killer]] Sudoku contains [[cage]]s and cage sums. Knowing the number of [[cell]]s in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the possible combinations of [[digit]]s that can go into the cage. For example, if we have a cage of 2 cells whose sum is 5, written 15[2], then either it contains the digits 1 and 4, or the digits 2 and 3. The cage combinations are written in set notation as {1,4} or {2,3}, or simply {14} or {23}.

	Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs\|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[~~elimination~~]]s, either simply because the candidate is not part of any combination, or in conjunction with some other [[technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.		Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs\|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[eliminate\|eliminations]], either simply because the candidate is not part of any combination, or in conjunction with some other [[solving technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.

	== Example 1 <English version>==		== Example 1 <English version>==

127.0.0.1: Created page with "A Killer Sudoku contains cages and cage sums. Knowing the number of cells in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the p..."

2021-10-24T14:46:31Z

Created page with "A Killer Sudoku contains cages and cage sums. Knowing the number of cells in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the p..."

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A [[Killer]] Sudoku contains [[cage]]s and cage sums. Knowing the number of [[cell]]s in a cage and its sum, we can determine the possible '''cage combinations''', i.e., the possible combinations of [[digit]]s that can go into the cage. For example, if we have a cage of 2 cells whose sum is 5, written 15[2], then either it contains the digits 1 and 4, or the digits 2 and 3. The cage combinations are written in set notation as {1,4} or {2,3}, or simply {14} or {23}.

Manually enumerating the cage combinations for arbitrary cages can be very tedious, but some [[Sudoku Programs|helper programs]] such as [[SumoCue]] can automatically compute the cage combinations for the player. However, for some cages, the cage combinations are easy to enumerate and can lead to [[candidate]] [[elimination]]s, either simply because the candidate is not part of any combination, or in conjunction with some other [[technique]]. A suggested way of enumerating the combinations is to start from the largest possible digit and then working systematically downwards, so a cage 11[3] would be enumerated as {8,2,1} or {7,3,1} or {6,4,1} or {6,3,2} or {5,4,2}. One should be careful to list all combinations.

== Example 1 <English version>==
For a cage 24[3], its only combination is {7,8,9}. Then we can eliminate the digits 1 to 6 from all three cells in the cage. Further, if all the three cells all belong to the same [[line]] or [[nonet]], we can then apply [[Naked Triple]] to that line or nonet.

== Пример 1 <Russian version>==
Для клетки 24[3], только комбинация - {7,8,9}. Затем мы можем устранить цифрам 1 на 6 из всех трех ячеек в клетке. Продвигать, если все три ячейки все принадлежат тому же [[line]] or [[nonet]], мы можем тогда приложите [[Обнаженный Тройной]] на эту линию или nonet.

== Example 2 <English version>==
Given a cage 28[4], there are two possible cage combinations: {4,7,8,9} or {5,6,8,9}. Either way, the cage must contain the digits 8 and 9 in 2 of the 4 cells. Knowing this, we might be able to apply the [[Locked Candidates]] technique. Further, if a cell whose candidates are 8 and 9 sees all 4 cells, we may be able to apply the [[Killer Subset]] technique.

== Пример 2 <Russian version>==
Данна клетка 28[4], есть две возможных комбинации клетки: {4,7,8,9} или {5,6,8,9}. Любой путь, клетка должна содержать цифрам 8 и 9 в 2 4 ячеек. Зная это, мы вероятно способными приложить [[Locked Candidates]](Запертые Кандидаты) техника. Продвигать, если ячейка, чьи кандидаты - 8 и 9 видит все 4 ячейки, мы возможно способными приложить [[Killer Subset]](Подмножество Убийц) техника.

== Example 3 <English version> ==
Suppose we have a cage 15[3] such that
* Cell A has candidates 3, 4, 5.
* Cell B has candidates 7, 8.
* Cell C has candidates 1, 2, 3, 4, 5, 6.

We could enumerate the permutations (3+7+5 or 3+8+4 or 4+8+3 or 5+7+3 or 5+8+2) and conclude that 1 and 6 can be eliminated from the cell C.

Alternatively, we can also consider the minimum and maximum possible values of the sum A + B. Since 3 <= A <= 5 and 7 <= B <= 8, we have 10 <= A + B <= 13. Due to the constraint A + B + C = 15, we have 2 <= C <= 5. This means that we can eliminate 1 and 6 from cell C.

== Пример 3 <Russian version> ==
Полагаем, что у нас есть клетка 15[3] так что
* Ячейка имеет кандидатов 3, 4, 5.
* Ячейка B имеет кандидатов 7, 8.
* Ячейка C имеет кандидатов 1, 2, 3, 4, 5, 6.

Мы могли бы перечислить перестановки (3+7+5 или 3+8+4 или 4+8+3 или 5+7+3 или 5+8+2) и решать, что 1 и 6 могут быть устранены из ячейки C.

Кроме того, мы можем также рассматривать минимальные и максимальные возможные величины суммы + B. С 3 <= <= 5 и 7 <= B <= 8, у нас есть 10 <= + B <= 13. Из-за ограничения + B + C = 15, у нас есть 2 <= C <= 5. Это означает, что мы можем устранить 1 и 6 из ячейки C.

Author: Email: Javof@mail.ru
ICQ:462675542

[[Category:Killer Sudoku]]
[[Category:Solving Techniques]]