The New Sudoku Players' Forum

[Myth Jellies]

Multi-spoke patterns are described here.

In the following, cells marked with '/' do not contain X, Y, or Z.

A simple triple spoke pattern is shown below.

Code: Select all

.---------------------.----------.----------.
| .      .      .     | .  /  / | /  /  /  |
| XYZ    .      .     | .  .  . | .  .  .  |
| .      .      .     | .  .  . | .  .  .  |
.---------------------.----------.----------.

(X)r1c123 ======================= (X)hub_r1c4
 |                                 |
(X = Y = Z)AALS_rim - (Z)r1c123 = (Z)hub_r1c4
     |                             |
    (Y)r1c123 =================== (Y)hub_r1c4


The hub is in r1c4. The three spokes for X, Y, and Z all follow r1c123. The AALS rim is in r2c1. This triple spoke eliminates digits other than X, Y, and Z from the hub. It also eliminates all X, Y, and Z from r2c23|r3c123

One way to expand this triple-spoke is to consider the Almost Locked Candidates (ALC) pattern shown below.

Code: Select all

.---------------------.---------.----------.
| .      .      .     | .  .  / | /  /  /  |
| XYZ    XYZ    .     | .  .  . | .  .  .  |
| .      .      .     | .  .  . | .  .  .  |
.---------------------.---------.----------.

The ALC net loop above has the form

 (XorY)r1c123 ===================== (X&Y)r1c45
  |                                   |
(X&Y=X&Z=Y&Z)r2c12 - (YorZ)r1c123 = (Y&Z)r1c45
      |                               |
     (XorZ)r1c123 ================= (X&Z)r1c45

The eliminations for the ALC pattern are nearly the same. The hub (r1c45) can only contain X, Y, and Z. Also, X, Y, and Z can be removed from r2c3|r3c123. You can probably see how the ALC is an extension of the triple-spoke pattern. Essentially, you buy an extra cell for the hub by reducing the rim from an AALS to an ALS

The triple-spoke can also expand as follows...

Code: Select all

.---------------------.---------.----------.
| .      .      .     | .  /  / | /  /  /  |
| WXYZ   .      .     | .  .  . | .  .  .  |
| WXYZ   .      .     | .  .  . | .  .  .  |
.---------------------.---------.----------.

     +--------------------------------------------------+
     |     +----------------------------+               |
     |     |                            |               |
  *(W&Z=XorY)AALS_rim     ----(Y)hub = (Y)r1c123 ----+  |
     |  |                /     |                     |  |
     | (X)r1c123 ===== (X)hub  |                 (ZorY=X&W)AALS_rim*
     |  |                \     |                  |     |
  *(W&Y=XorZ)AALS_rim     ----(Z)hub = (Z)r1c123 -+     |
     |     |                            |               |
     |     +----------------------------+               |
     +--------------------------------------------------+


The AALS rim here is r23c1 and the hub is in r1c4 again. Noting that the '/' cells can contain W's, the closed net above results in the following deductions. r1c4 = XYZ, r23c1 contains W, and r23c23 <> XYZ

A similar expansion is available for the ALC case.

Code: Select all

.---------------------.---------.----------.
| .      .      .     | .  .  / | /  /  /  |
| WXYZ   WXYZ   WXYZ  | .  .  . | .  .  .  |
| .      .      .     | .  .  . | .  .  .  |
.---------------------.---------.----------.

With similar deductions.

[Steve K]

Hi Myth!
One of those examples looks awfully familiar!
I had been trying to post similar expansions of the concept at your other similar thread. However, the insight you provide regarding the interaction between degrees of freedom is fantastic. It also yields readily to a general theorem describing this pattern.