I think I've come up with a good way to explain how

to use type-3 unique rectangles to eliminate candidates.

So, I'd like to present alternative explanations for

2 examples presented earlier. I apologize in advance

if I am wrong and the explanations are redundant

or otherwise not useful.

Here's an example from an

earlier post:- Code: Select all
`+----------------+----------------+----------------+`

| 39 4 1 | 2 6 5 | 389 378 379 |

| 5 7 6 | 3 9 8 | 4 1 2 |

| 2389 389 289 | 7 1 4 | 6 5 39 |

+----------------+----------------+----------------+

| 3478 6 478 | 5 2 9 | 1 378 347 |

| 1 39 279 | 4 8 6 | 239 237 5 |

| 2489 89 5 | 1 7 3 | 289 6 49 |

+----------------+----------------+----------------+

| 89 5 89 | 6 4 7 | 23 23 1 |

| 6 2 3 | 9 5 1 | 7 4 8 |

| 47 1 47 | 8 3 2 | 5 9 6 |

+----------------+----------------+----------------+

The floor cells are r9c1 and r9c3, and the roof cells

are r4c1 and r4c3. 4 and 7 are candidates in all of the

roof cells and floor cells. The extra candidates in the

roof cells are 3 and 8.

Assuming the puzzle has a unique solution, there

are 2 constraints on the roof cells:

1) At least 1 of the values 3 and 8 must go in 1 of the roof cells.

2) At least 1 of values 4 and 7 must go in a cell different than the roof cells, in each group that contains the roof cells.

Each of these constraints might lead to eliminating candidates.

Let's start with constraint 1, and combine it with other candidate information in box 4 to get the following:

- At least 1 of the values 3 and 8 must go in either r4c1 or r4c3

- 1 of the values 3 and 9 must go in r5c2

- 1 of the values 8 and 9 must go in r6c2

Combining this, the 3 values 3,8, and 9 must go

somewhere in the 4 cells r4c1, r4c3, r5c2, and r6c2.

So we can remove these 3 values as candidates from all

other cells in the box (we can remove an 8 and 9

from r6c1 and a 9 from r5c3).

The logic used is of course similar to that used

for a naked triple. But using that term in this

context might be confusing because 1) there

are 4 cells, and 2) some of the cells contain candidates

other than 3,8, and 9.

Let's reset the puzzle back to the candidates shown above, and

try to use constraint 2 instead of constraint 1.

- Code: Select all
`+----------------+----------------+----------------+`

| 39 4 1 | 2 6 5 | 389 378 379 |

| 5 7 6 | 3 9 8 | 4 1 2 |

| 2389 389 289 | 7 1 4 | 6 5 39 |

+----------------+----------------+----------------+

| 3478 6 478 | 5 2 9 | 1 378 347 |

| 1 39 279 | 4 8 6 | 239 237 5 |

| 2489 89 5 | 1 7 3 | 289 6 49 |

+----------------+----------------+----------------+

| 89 5 89 | 6 4 7 | 23 23 1 |

| 6 2 3 | 9 5 1 | 7 4 8 |

| 47 1 47 | 8 3 2 | 5 9 6 |

+----------------+----------------+----------------+

The key is to look for all other cells (besides the

roof cells) that contain a 4 or 7 candidate. We find

cells r6c1 and r5c3.

By constraint 2, we know that at least 1 of the values 4 and 7 must

go into 1 of these 2 cells. By the listed candidates,

we see that a 2 must also go into 1 of these cells.

So, in the end, the two cells r6c1 and r5c3 are going

to contain a 2 and a 4 or 7, and we can eliminate

all other candidates from these 2 cells.

Using constraint 2 leads to the same result as using

constraint 1, in this case.

Here's one more example from

this post:- Code: Select all
`+----------------+----------------+----------------+`

| 457 39 39 | 57 2 1 | 6 457 8 |

| 157 8 2 | 57 4 6 | 1579 579 3 |

| 1457 467 16 | 9 3 8 | 1457 2457 257 |

+----------------+----------------+----------------+

| 2 346 136 | 8 9 7 | 45 45 16 |

| 478 4679 69 | 1 5 2 | 789 3 679 |

| 178 79 5 | 4 6 3 | 2 789 179 |

+----------------+----------------+----------------+

| 3 *25 4 | 6 8 9 | 57 1 *257 |

| 9 1 8 | 2 7 5 | 3 6 4 |

| 6 *25 7 | 3 1 4 | 589 2589*259 |

+----------------+----------------+----------------+

The cells that define the unique rectangle are marked with asterisks.

The floor cells are r7c2 and r9c2, and the roof cells

are r7c9 and r9c9. The extra candidates in the

roof cells are 7 and 9.

The 2 constraints on the roof cells are:

1) At least 1 of the values 7 and 9 must go in 1 of the roof cells.

2) At least 1 of the values 2 and 5 must go in a cell different

than the roof cells, in each group that

contains the roof cells.

Using constraint 1 and the listed candidates in column 9 we get:

- At least 1 of the values 7 and 9 must go in either r7c9 or r9c9

- 1 of the values 1,7, and 9 must go in r6c9

- 1 of the values 6,7, and 9 must go in r5c9

- 1 of the values 1 and 6 must go in r4c9

Combining this, the 4 values 1,6,7, and 9 must go

somewhere in the 5 cells listed above.

So we can remove these 4 values as candidates from all

other cells in column 9. (Here we can only remove

the 7 from r3c9).

Let's consider the same initial puzzle state again but this

time use constraint 2 instead.

- Code: Select all
`+----------------+----------------+----------------+`

| 457 39 39 | 57 2 1 | 6 457 8 |

| 157 8 2 | 57 4 6 | 1579 579 3 |

| 1457 467 16 | 9 3 8 | 1457 2457 257 |

+----------------+----------------+----------------+

| 2 346 136 | 8 9 7 | 45 45 16 |

| 478 4679 69 | 1 5 2 | 789 3 679 |

| 178 79 5 | 4 6 3 | 2 789 179 |

+----------------+----------------+----------------+

| 3 *25 4 | 6 8 9 | 57 1 *257 |

| 9 1 8 | 2 7 5 | 3 6 4 |

| 6 *25 7 | 3 1 4 | 589 2589*259 |

+----------------+----------------+----------------+

We look for all cells (besides the roof cells) containing

a 2 or 5 candidate. We find only r3c9. By constraint 2,

a 2 or a 5 must go into this cell in the end. So we can remove

7 as a candidate there.

Using constraint 2 again leads to the same result as using

constraint 1. (I'm not sure: will this always be true?)