This

diamond constraint is found in several "restrained prototypes" which HATMAN and I have been studying. It reminds me of the "

DG diagonals" that HATMAN introduced to me many months ago.

Just to recap, these are the normal diagonals:

- Code: Select all
`A..|...|..B`

.A.|...|.B.

..A|...|B..

---+---+---

...|A.B|...

...|.X.|...

...|B.A|...

---+---+---

..B|...|A..

.B.|...|.A.

B..|...|..A

These are the DG diagonals:

- Code: Select all
`A..|.A.|..A`

...|...|...

..B|.B.|B..

---+---+---

...|...|...

A.B|.X.|B.A

...|...|...

---+---+---

..B|.B.|B..

...|...|...

A..|.A.|..A

If you read the cells row by row and list out the column indices, a leading diagonal would have the list [123456789] and a non-leading one would have [987654321]. As for the DG ones, you're reading the cells box by box and list out the "position indices" (within each box), and the lists are also [123456789] & [987654321] respectively.

Now look at the "diamond":

- Code: Select all
`A..|...|...`

...|A..|...

...|...|A..

---+---+---

.A.|...|...

...|.A.|...

...|...|.A.

---+---+---

..A|...|...

...|..A|...

...|...|..A

If you list out the column indices row by row, the list is [147258369]. Now if you list out the position indices box by box, it is [147258369] again! And for completeness we should have a "non-leading diamond" to complement this "leading diamond", with the list [963852741]:

- Code: Select all
`...|...|..B`

...|..B|...

..B|...|...

---+---+---

...|...|.B.

...|.B.|...

.B.|...|...

---+---+---

...|...|B..

...|B..|...

B..|...|...

So, for the sake of elegancy we should focus on puzzles with both "diamonds" instead of those with only one, just as we usually deal with X puzzles instead of \ or / puzzles.

Now here is a solution grid of a puzzle with both diamonds, both diagonals, both DG diagonals, the DG property and the additional properties that each mini-row sums to 15 and each mini-column being a set of 3 consecutive digits (and it even includes all 18 "generalised diamonds" as proposed by Bill):

159 672 834

267 483 915

348 591 726

726 348 591

834 159 672

915 267 483

483 915 267

591 726 348

672 834 159

Of all the "restrained prototypes" I know, only some NNC (kNight Non-Consecutive) grids has both diamonds, e.g.:

231 564 897

456 789 123

978 312 645

897 231 564

123 456 789

645 978 312

564 897 231

789 123 456

312 645 978

Also, in "NC AK AN" and "FNC AK AN" grids, one of the diamonds has 1..9 while the other is solely occupied by a single digit. In "FNC CNC" grids, both diamonds are occupied by 3 digits 3 times each. In "FNC AK X" grids, none of the diamonds are neatly occupied (e.g. one has [117654399], one has [438555276]).

PS:

tarek, don't think your "

Zorro" constraint is of any particular significance other than fictional value, just like we don't study this constraint much:

- Code: Select all
`A..|...|...`

.A.|...|...

..A|...|...

---+---+---

...|..A|...

...|.A.|...

...|A..|...

---+---+---

...|...|A..

...|...|.A.

...|...|..A