Coloring can be combined with freeform pattern analysis to determine a small set of pattern trials for Heine’s ultrahardcore 1. Typical choices for this decision are reviewed. A single trial produces the solution. For this we go back to the starting grid of the previous post, just after the simple 1-way of August 3.
Here, a pattern is a set of candidates of a single value with one member in each row and column having no placement of that value. Given a cluster of that value, the true pattern can include candidates of only one of the cluster colors. This fact can be used to limit possible patterns for trials. Looking at clusters on the UHC 1 panels, . . .
In the North to South freeform patterns of 7, we have two compatible with green and two compatible with blue.
A coloring trial of blue will use two placed 7-cells, but a trial of one of the two blue patterns will use 6 7-cells, and will be much more decisive. On the other hand, the failure will not determine color, and will leave up to two more failures before pattern and color are known.
In the 4-panel, North to South, we get one red pattern and four orange.
In this case, the red trial enlists 5 cells determines color, but failure leaves four orange patterns to be tested.
Looking further, the 5-panel doesn’t have slinks interposed in two directions, and therefore imposes no color restrictions on the number of patterns.
The better trial plan comes from the 6- panel with a cluster of 6 cells, also having two line slinks in each direction. The only feasible direction is East to West, and there is one pattern compatible with each color.
A failure of one trial guarantees the success of the other.
Before we explore the trial of one of these patterns, consider that the Single Alternate Sue de Coq of the previous post is also a very decisive trial. This leads us to ask, how far would we have to take that trial to know which of these two patterns is true? The answer is striking. In the trace of the initial follow up, before the double hidden unique rectangle result, W56 of the trace marks the naked pair that reveals the red pattern to be false. We could add the orange pattern to the SASdC trial, making it more decisive, but it’s still a trial. We can’t add the (NW3, NW4) of the SASdC trial to the orange pattern trial until we know that the pattern implies these two placements.
Here is the set up for the orange pattern trial. Knowing from the trial of the last post that the red pattern is false, let’s see if the orange pattern trial reaches the solution, how it compares to the SASdC trial.
The pattern follow up leaves a naked quad in r7 and a 2-chain ANL.
The ANL removal brings another quad in c2.
Followed by an effective ANL.
And after E2 =>SE1=>SE2, a naked triple and Sr7 boxline,
a simple AIC places NW4, removing 4r8c1 and removing 13r8c3. Then right next door, a hidden unique rectangle removes 4r9c4. Trace what happens if 4r9c4 is true.
After S4 => SW2, we need two more colors for the extensive cluster. The trap expands blue green enough to remove 9r6c6, and rouge/tan enough to remove 1r13c6.
Then a simple ANL removal allows
A red wrapping ANL that happens to trap 3r1c6 as well, and tan takes over
The pattern trial was easier, but it could have required two of them.
Next time, we continue 0n left side with ultrahardcore 1 + 44 = 45.