## Anti Chess, NC and Forbidden Pairs (FP) puzzles

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

I think I still don't fully understand your rules:
No adjacent cells can sum, multiply, divide or subtract to give you 5 or 10
Toroidal grid: The grid displays a top-bottom right-left wrapping to
give you a toroidal (doughnut) shape

And look at this board state, which steps do solve this puzzle?
Code: Select all
`+-----------+-----------+-----------+|  11 4  3  |  1  8  7  |  6  2  9  ||  9  2  6  |  3  4  11 |  8  1  7  ||  8  1  7  |  6  2  9  |  . .  .  |+-----------+-----------+-----------+|  1  7  8  |  9  6  2  |  .  . .  ||  2  6  9  |  11 3  4  |  7  8  1  ||  4  3  11 |  7  1  8  |  9  6  2  |+-----------+-----------+-----------+|  3  11 4  |  8  7  1  |  2  9  6  ||  6  9  2  |  4  11 3  |  1  7  8  ||  7  8  1  |  2  9  6  |  3  4  11 |+-----------+-----------+-----------+`

Do you mean with adjacent all 8 surrounding, or 4 with + or 4 with x?
And the math operators, on which cell are they used?
And 1 operator for the target adjacent cells?
creint

Posts: 230
Joined: 20 January 2018

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

Hi creint,

I'm following the Sudoku federation terminology regarding the words "Adjacent" and "Neighbouring":

Adjacent cell means any cell that shares an edge. That means on our rectangular grid with square cells that each cell can have a maximum of 4 adjacent cells (1 in each orthogonal direction). On a toroidal grid each cell will have 4 adjacent cells. On a grid without a continuity plane (non toroidal) then the cells at the edge of the board will have less adjacent cells and the corner cells will have only 2 adjacent cells each.

Neighbouring cell means any cell that shares an edge or a point. That means on our rectangular grid with square cells that each cell can have a maximum of 8 neighbouring cells (4 adjacent cells in each orthogonal direction and 4 cells sharing a point in each diagonal direction). On a toroidal grid each cell will have 8 neighbouring cells. On a grid without a continuity plane (non toroidal) then the cells at the edge of the board will have less neighbouring cells and the corner cells will have only 3 neighbouring cells each.

with the no XV puzzle The rules state that we don't have 5 nor 10 as clues and that no adjacent cells can give you 5 or 10 through arithmetic operations. The possibilities therefore are:
Code: Select all
`1 + 4 = 51 + 9 = 102 + 3 = 52 + 8 = 103 + 7 = 104 + 6 = 106 - 1 = 57 - 2 = 58 - 3 = 59 - 4 = 511- 1 = 1011- 6 = 5`
Each of these pairing is a forbidden pair and therefore can't occupy adjacent cells which should make solving those empty cells straightforwrad

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

tarek wrote:with the no XV puzzle The rules state that we don't have 5 nor 10 as clues and that no adjacent cells can give you 5 or 10 through arithmetic operations. The possibilities therefore are:
Code: Select all
`1 + 4 = 51 + 9 = 102 + 3 = 52 + 8 = 103 + 7 = 104 + 6 = 106 - 1 = 57 - 2 = 58 - 3 = 59 - 4 = 511- 1 = 1011- 6 = 5`
Each of these pairing is a forbidden pair and therefore can't occupy adjacent cells which should make solving those empty cells straightforwrad

If the definition is "arithmetic operations" then the list should include 1 x 5 = 5, and 2 x 5 = 10, shouldn't it ?

Mathimagics
2017 Supporter

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### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

Mathimagics wrote:If the definition is "arithmetic operations" then the list should include 1 x 5 = 5, and 2 x 5 = 10, shouldn't it ?
the digit 5 is not there as clue as per the instructions. That is why it is not included. The clues are 1,2,3,4,6,7,8,9,11 we have therefore 11 - 6 = 5 & 11 - 1 = 10 as possibilities and therefore should be forbidden pairs.

The idea is to:
Remove 5 to have 1,2,3,4,6,7,8,9,10
Remove 10 to have 1,2,3,4,6,7,8,11 to make it a "no XV" in both clues and arithmetic operations

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

Yes it solves now.
In the picture:
Adjacent cells can't combine to give 5 or 5 through arithmetic operations

Which was in my opinion something like operations on the four cells: a+b+c+d not equal 5 or 10.
The clue that was missing for me was always 2 cells and center cell + one of the four adjacent.
creint

Posts: 230
Joined: 20 January 2018

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

creint wrote:Which was in my opinion something like operations on the four cells: a+b+c+d not equal 5 or 10.
The clue that was missing for me was always 2 cells and center cell + one of the four adjacent.

Thanks, I agree that the wording needs to be as clear as possible. A possible better option is "adjacent cells can't give 5 or 10 through arithmetic operations" or "adjacent cells can't give 5 or 10 through +,-,x,/". At the moment I think that "adjacent cells" should mean "any 2 adjacent cells" and therfore didn't elaborate on that further!

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

Hello, do you know how many Anti-Knight sudokus puzzle there are ? difficult question for me ....(just Anti-Knight)
JFA

Posts: 4
Joined: 08 April 2019

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

JFA wrote:Hello, do you know how many Anti-Knight sudokus puzzle there are ? difficult question for me ....(just Anti-Knight)
Apologies for a late reply. I can see the the question was addressed in a different thread to a satisfactory level.

tarek

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

Ordered NC Vanilla 1

I always find that NC is too restrained and CNC even more so, hence when trying to make a killer the cages end up too large. They also cause what I call the cliff-edge problem: add a clue to make the puzzle solvable and then it is too easy. This is particularly the case with MeanDoku.

I therefore thought to reduce the restraint by requiring NC in one direction only. I have gone for rejecting increasing as they are slightly more obvious visually. Due to reduced interactions the restraint level is reduced by more than 50%, I would guess to about one third (the number crunchers among you may wish to calculate the number of solutions).

Ordered (orthogonal) NC: horizontally and vertical increasing consecutive numbers are not allowed so 9-8 is OK but 8-9 is not.

tarek has kindly provided me with a set of solutions so here is the first one.

Probably medium difficulty once you are used to it, as it took me a few tries to do it without fishes.

HATMAN

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Location: Nigeria

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

tarek et al

The concept of 50% as restrained is a very human analysis. If we wanted to look at it in terms of number of solutions what sort of relationship would we use geometric, exponential or some other?

Maurice
HATMAN

Posts: 272
Joined: 25 February 2006
Location: Nigeria

### Re: Ordered NC

HATMAN wrote:I therefore thought to reduce the restraint by requiring NC in one direction only. I have gone for rejecting increasing as they are slightly more obvious visually. Due to reduced interactions the restraint level is reduced by more than 50%, I would guess to about one third (the number crunchers among you may wish to calculate the number of solutions).

Ordered (orthogonal) NC: horizontally and vertical increasing consecutive numbers are not allowed so 9-8 is OK but 8-9 is not.

This variant is like a cloth "nap" with a direction to it. There are some symmetry operations that work with it but the rest would invalidate it.

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

Minimal Restraints K0

Recent posts have been complex and computer solution driven, so naturally I'm reverting to P&P.

I'm not sure if these puzzles are interesting enough to continue with so I've put a batch of Killers on the assassin site for peoples views.

When a Restrained Killer Sudoku is created in JSudoku you can invariably solve it with very little use of the constraint. In Vanilla's that is not the case. I'm not sure if this is a killer point or Jean-Christophe's coding (any thoughts?). Hence I thought why apply the constraint everywhere, why not just apply it to a few cells, perhaps unknown.

Where you choose a cell that choice may not be unique but the solution is. For those where I have given placement they obviously can be solved without knowing them. In these cases I believe the solution is unique but do not have the tools to prove that without heavy labour.

I have tried creating a Vanilla (MC 3) and I still have not finished creating it, so I am taking a different approach: see snowflakes in the next post.[/size]

MR Killer 0 NC 1-cell
This is an easy one to start.
Find one cell that is NC.

HATMAN

Posts: 272
Joined: 25 February 2006
Location: Nigeria

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

MR Snowflakes 1&2

I've been intrigued by combined restraints for a long while; these are kind of maximal. A snowflake cell is AK, AN, NC, FNC and KnNC.

MR Snowflake 1 plus NC

The yellow givens are snowflakes. The two fives are just NC.

MR Snowflake 2 plus NC

The yellow givens are snowflakes. The two blue cells are just NC.

HATMAN

Posts: 272
Joined: 25 February 2006
Location: Nigeria

### Re: Anti Chess, NC and Forbidden Pairs (FP) puzzles

HATMAN wrote:I always find that NC is too restrained and CNC even more so, hence when trying to make a killer the cages end up too large.

I've seen in some cases the use of NC-x, which means the difference between them can't be x. Those are less restrained than the default NC-1.

Code: Select all
`Eliminations    NC-1  NC-2  NC-3  NC-4  NC-5  NC-6  NC-7  NC-81    2     3     4     5    6     7     8     92   13     4     5     6    7     8     9     -3   24    15     6     7    8     9     -     -4   35    26    17     8    9     -     -     -5   46    37    28    19    -     -     -     -6   57    48    39    2     1     -     -     -7   68    59    4     3     2     1     -     -8   79    6     5     4     3     2     1     -9   8     7     6     5     4     3     2     1`

You could also have NC<x but that would make it more restricted instead. Another variation on that is the "Sum less than x"

Code: Select all
`Eliminations    SLT-11  ...  SLT-14  ...  SLT-171   -            -            -2   9            -            -3   89           -            -4   789          -            -5   6789         9            -6   56789        89           -7   456789       789          -8   3456789      6789         99   23456789     56789        89`

Others:

- No-XV - Sum can't be 5 or 10
- No-XV(t) - No adjacent cells can sum, multiply, divide or subtract to give you 5 or 10 (see terek's post in the previous page)
- No-Prime - Sum can't be a prime
- No-M3 - Sum can't be multiple of 3

Code: Select all
`Eliminations    No-XV   No-XV(t)  No-Prime  No-M31   49      49-5      246       2582   38      38-5      1359      1473   27      27         248       3694   16      16         1379      2585    5       5-12       268      1476    4      14          157       3697    3      23           46       2588    2      32           359      1479    1      41           248       369`
Wecoc

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