Serg wrote:what grids are you choosing for new bands' production? Is this random choice (from known grids having rank > RMIN)?

It started with a pool of just 50,000 grids with rank > 16500 (the result of months of previous searching). Every grid in this pool was processed in the first round, and every new grid in the 2nd round, etc.

At the end of each round, the pool is expanded further by processing the new grids with the existing (UA) vicinity search method. After just 4 rounds (2 weeks) the total pool size has grown to 650,000.

When the pool becomes too big, then I could introduce random selection, but I rather think it might be simpler just to increase RMIN from 16500 to some higher value.

Serg wrote:It looks like we should replace band/stack with lowest rank by new band/stack with rank >= Rn, isn't it?

That would be ideal, yes, but we can't easily perform that operation: "replace band Rm by new band with rank >= Rn".

We

can, however, go through ALL of the bands that can replace Rm (effectively, this means getting all solutions to the corresponding puzzle with 3 bands fixed) and check each case for the desired properties.

Consider a generalised 3-band puzzle, where we have removed one band, then transposed the grid so is a "3-stack puzzle". The boxes are labelled as band/stack numbers:

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

| 11 | 12 | 13 | |

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

| 21 | 22 | 23 | |

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

| 31 | 32 | 33 | |

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

| 41 | 42 | 34 | |

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

Suppose that stacks 1 to 3 have rank >= X. We can easily enumerate recursively all the possible Box(14) settings that result in a band 1 with rank >= X.

For each of these Box(14) settings, we can then apply the same recursive process to Box(24), and so on for Box(34) and Box(44).

But we still have to check every valid Box(44) completion, looking at rank of stack 4, since the stack rank can't really be determined until it is completed.

(In practice, it is actually faster to stop the recursion after Box(14), testing each case with a fast solver to generate all the solutions for the remaining 3 boxes.)

After the 4th round (of 1 band removals + UA vicinity search) the search status is now:

- we have found grids for 17,000 different ranks. This includes all ranks in the range [0, 16926]. The maximum rank for which a grid example has been found is still 17115.