LinkedIn Patches #35 Answer

🧩 Deep Logic Analysis

Solving Patches #35 requires a disciplined practice of identifying "anchor shapes"—those with constraints so rigid they leave no room for error.

1. The Yellow 9 Square: The $3 \times 3$ Yellow Square is the gravitational center of this puzzle. In a $5 \times 5$ grid, a $3 \times 3$ shape is massive. Given its position in the bottom-left quadrant, it must occupy the area from Row 3 to Row 5 (vertically) and Column 1 to Column 3 (horizontally). However, since there is a Green 4 and a Grey 2 clue on the bottom row, the Yellow 9 is forced upward, occupying Rows 2, 3, and 4.
2. The Perimeter 5s: We have two clues for the number 5 (Light Blue and Purple) located in the top corners. Because they are 5 units long in a $5 \times 5$ grid, they must span an entire edge. To avoid collision, the Light Blue 5 claims the entire top row (Row 1), while the Purple 5 takes the entire right-most column (Column 5).
3. The Orange 6 Tall Rectangle: With the top row and right column occupied, the Orange 6 (a $3 \times 2$ vertical rectangle) has only one logical home. It must sit adjacent to the Purple 5, filling the remaining height in the middle-right section.
4. The Bottom Row Sequence: The Green 4 and Grey 2 share the bottom row. Since the Grey 2 is in the corner and the Green 4 must be $1 \times 4$ or $2 \times 2$, the only way to fit both is a $1 \times 4$ horizontal strip for Green and a $1 \times 2$ horizontal strip for Grey (overlapping at the corner).
5. Closing the Gaps: This leaves small gaps for the Teal and Red shapes. The Teal "Wide" clue fills the $1 \times 3$ void between the Light Blue top and the Yellow square, while the Red "Wide" clue fits perfectly into the $1 \times 2$ slot between the Yellow square and the Orange rectangle.

🎓 Lessons Learned From Patches #35

• The Square Anchor Strategy: Whenever you see a large square (like the Yellow 9), place it first. Its high area-to-grid ratio severely limits the "legal" moves for surrounding pieces, often solving 30-40% of the grid instantly.
• Edge Exclusion: In a $5 \times 5$ grid, any clue with the number 5 is a "Full-Length" hint. Identifying whether it is a row or a column depends on the neighboring corner clues. If two 5s meet at a corner, one must be horizontal and one must be vertical.

💡 Trivia

• The Magic of 9: In Patches, the number 9 is the largest possible "Perfect Square" clue that can appear in a $5 \times 5$ grid while still allowing other shapes to exist. Mathematically, a $4 \times 4$ (16) would leave almost no room for the mandatory variety of LinkedIn's logic puzzles.
• Tiling Theory: This puzzle is a variation of "Squared Square" tiling, a mathematical challenge first solved in 1925. While our patches aren't all squares, the logic of fitting rectangles into a finite space without gaps is a fundamental problem in geometric combinatorics.

❓ FAQ

Can the Light Blue 5 be a vertical column instead of a horizontal row?
No. If the Light Blue 5 were a vertical column on the left, it would overlap with the Green 4 and the Yellow 9's required footprint, creating a logical contradiction that breaks the grid.

Why is the Green shape a 1x4 strip and not a 2x2 square?
The Green clue features a "Square/Cross" icon, which allows for multiple configurations, but the available space on the bottom row is limited by the Grey 2. A $2 \times 2$ Green square would block the Yellow 9 from completing its $3 \times 3$ dimensions.

What determines the size of the Teal and Red patches if they have no numbers?
In Patches, these are "Filler Clues." Their dimensions are dictated entirely by the "Negative Space" left behind by the numbered anchors. You must solve the 9, 6, and 5s first to reveal the exact boundaries for the Teal and Red rectangles.