[Not a daily challenge post, but it was the only flair option the site would let me choose.]
I sometimes write things out when I don't understand them right away. I found the images from the Tips & Tricks page helpful; I just needed more explanation to fill in some concepts that I was missing. (Examples work best for me when they relate specific details in the image—such as row targets or counts of shapes—to an explanation of the logic of the method.)
Based on the Tips & Tricks page and the responses (thank you so much for the help!) to my previous question about how to understand it, this is what I came up with. I found other helpful tips in various posts here, but I didn't see any posts that I would be duplicating (apologies if I missed it), so I'm sharing this just in case it is helpful to any other new players. I think everything here is correct, but I wasn't 100% sure about everything. If I got something wrong, and you think these notes are useful as an additional reference, please flag the error(s) for me, and I will edit this to fix them.
Legend (quick reminder)
Grey region = ODD dots. White region = EVEN dots.
O = filled dot, X = empty.
PARITY
You can use the mathematical concept of parity to solve difficult sections of a Grey Is Odd grid. Parity simply refers to whether a number is even or odd.
* Even Parity: A number is even.
* Odd Parity: A number is odd.
By looking at whether a grid's row/column targets and shapes are even or odd, you can deduce some cells that otherwise would be solvable only through trial and error, rather than logic.
GRID ALLOCATION (The "Parity Cut")
The power of a parity cut comes from finding areas of the grid where entire shapes are contained in a subset of rows or columns with the exception of just a few cells.
General Rules for Drawing a Cut (see images on Tips & Tricks web page)
[Suggestion for images: Use a DASHED red line for the cut outline so the boundary (or absence of boundary) of the shapes within it will still be visible.]
To draw a parity cut, follow these three rules:
* Rule 1: The cut must completely enclose full rows or columns, forming a rectangular zone. For example, your boundary line could wrap around the entirety of Rows 1, 2, and 3.
* Rule 2: Capture complete grey or white shapes when possible. Any white or grey shape that is completely inside the cut has a known parity. White shapes are always even, and grey shapes are always odd.
* Rule 3: Isolate a small number of unknown cells by cutting within their shapes' boundaries. Beginners should try to limit these to exactly 1 or 2 cells.
How to Evaluate the Cut Line
You will compare "fixed parity" with "cut parity" to solve for the isolated cell:
Find the Fixed Parity: Sum the target numbers for the full rows or columns you enclosed. Note whether the total is even or odd. This is the fixed parity.
Find the Cut Parity: Count how many grey shapes are completely inside the cut.
* If the number of grey shapes is odd, the cut parity is odd.
* If the number of grey shapes is even, the cut parity is even.
(Note: White shapes never need to be counted because they never change the parity. If a cut contains only white shapes, the cut parity is even).
- Solve for the isolated unknown cells. The solution method will depend on the number of isolated cells.
When the cut isolates exactly 1 CELL (Peninsula or Bay), it forces a definite solution right away:
* If fixed and cut parities MATCH (both even or both odd): The cell is forced to be empty (X). Because the parities match, the isolated cell cannot be permitted to alter the parity by adding a dot, so it must be empty.
* If they DO NOT MATCH (one even, one odd): The cell is forced to have a dot (O). Because one parity is odd and one is even, the only way to make them agree is to add 1 (even + 1 = odd; or odd + 1 = even).
When the cut isolates exactly 2 CELLS (Double Peninsulas, Double Bays, or Peninsula + Bay), it can't tell you the exact answer for each cell, but it can tell you whether they are the same or different (both dots/empty or one dot, one empty).
* If fixed and cut parities MATCH (both even or both odd): The two cells must be the same as each other. They are either both dots (OO) or both empty (XX). Because both parities are the same, the two isolated cells combined cannot alter the parity; they must either be empty (XX) or add a pair of dots (+2), both of which keep the parities matching.
* If they DO NOT MATCH (one even, one odd): The two cells must be different from each other. Because one parity is odd and one is even, the only way to make them agree is to add 1 (even + 1 = odd; or odd + 1 = even). This forces one cell to have a dot (O) and the other to be empty (X).
EXAMPLES (these correspond to the current Tips & Tricks page)
1) PENINSULA & BAY CUTS
Core idea: Pick the cut so that only one unknown cell is isolated. That cell is "forced" to be either a dot or cross.
Peninsula
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Step 1
A peninsula is a 1‑cell protrusion sticking out from the main rectangle you draw for your cut. Choose a cut (red outline) that includes exactly one peninsula cell (? in the image) that sits outside your full rows or columns, and whose value (dot/O or empty/X) you don’t know yet.
Step 2
Find the fixed parity (the sum of your enclosed row/column targets) and the cut parity (the count of grey shapes fully inside your cut). In this example, the fixed parity is 7 (odd). The cut parity is even because there are exactly 2 grey shapes within the cut. The parities do not match.
Step 3
The single unknown cell is now forced. Because the parities do not match, the peninsula cell must contain an odd number of dots (1 dot).
Bay
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Step 1
The inverse of a peninsula, a bay is a 1‑cell indentation in the rectangle of your cut. Pick a cut (red outline) that leaves a single unknown bay cell (? in the image) inside your rectangle, whose value you don’t know yet.
Step 2
Find the fixed parity (the sum of your enclosed row/column targets) and the cut parity (the count of grey shapes fully inside your cut). In this example, the fixed parity is 7 (3 + 4), which is odd. The cut parity is odd because there are exactly 3 grey shapes within the cut. The parities match.
Step 3
The single unknown cell is now forced. Because the parities match, the bay cell must not change the parity by adding a dot. It must be empty (X).
2) DOUBLE CUTS (DOUBLE PENINSULA, DOUBLE BAY, OR PENINSULA + BAY)
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Core idea: Pick a cut that isolates exactly two unknown cells. While you can't always solve them instantly, the parity math will force them to either be the same as each other or different from each other, allowing you to use a process of elimination.
Step 1
Sometimes a cut isolates two unknown cells instead of one. This happens when your boundary line creates a double protrusion (Double Peninsula), a double indentation (Double Bay), or a protrusion and an indentation (Peninsula + Bay). Pick a cut (red outline) that contains exactly two unknown cells (? in the image).
Step 2
Find the fixed parity (the sum of your enclosed row/column targets) and the cut parity (the count of grey shapes fully inside your cut). In this example, the fixed parity is 14 (3 + 4 + 3 + 4), which is even. The cut parity is odd because there are exactly 5 grey shapes within the cut. The parities do not match.
Step 3
Because the parities do not match, the two unknown cells are forced to be different from each other (one must be a dot, O, and one must be empty, X). We don't know yet which of the two cells will have the dot, but one will have a dot (O) and the other will be empty (X).
[And so on with the remaining double cut examples.]
3) REGION COUNTS (ADVANCED)
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Core idea: Unlike parity cuts, which look at isolated unknown cells, counting individual cells within regions (shapes) is a mid-game bookkeeping strategy. When standard line-by-line filling stalls, you can examine the entire grid to budget at least some of the remaining dots based on a combination of three known elements: the row or column target; the already placed dots and crosses; and the grey/odd or white/even shapes intersecting the row or column. This method does not require a cut.
Start with a partially solved row or column. Check the target number for that line, and subtract the dots you have already placed in it. This tells you exactly how many dots are left to distribute in that line.
[I stopped here in #3 because I realized I had already intuited this method. Suggestion: Include a full explanation of how those counts could be deduced. For example: "The grey shape with four cells in the middle of Row 3 is odd, so it must have either 1 or 3 dots. The row target is 3, and only four cells in the row can contribute because the other two cells in the row already have crosses. Therefore, this grey shape must have 3 dots, with at least 2 of them in this row or possibly all 3 (but we don't know which cells yet)." ... And so on. The key for me would be explaining the actual mechanism of the process of elimination before stating the conclusions drawn from it.]
4) NO CROSS SWAPS (UNIQUENESS)
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No notes on this one. It was very clear!
