Today's LinkedIn Patches #87 Answer

🧩 Deep Logic Analysis

To solve Patches #87, we must look at the constraints of the $7 \times 7$ grid. The logic flows from the most restricted areas—the corners and the long strips—inward to the center.

1. The Top-Down Anchor
The Yellow 6 at the top center is the primary pivot. Because it is positioned in the first row, a vertical $2 \times 3$ or $3 \times 2$ would prematurely block the middle columns. Instead, it forms a $1 \times 6$ horizontal strip. This forces the Orange 4 in the top-right corner to become a $1 \times 4$ vertical column, as there is only one column of width remaining on that edge.

2. The Eastern Wall
With the Orange 4 taking a vertical path, the Red 3 is squeezed. To avoid overlapping with the Blue 6 or the Pink 6, it must also adopt a vertical orientation, specifically a $1 \times 3$ strip. This "stacking" of vertical shapes on the right side is a classic Patches maneuver that clears the center for more complex blocks.

3. The Central Squeeze
The Green 6 sits in the dead center. Given the vertical walls we built on the right, the only way to accommodate 6 tiles without colliding with the Teal 4 or Blue 6 is a $1 \times 6$ horizontal bridge. Once this is placed, the Teal 4 is pushed into a $2 \times 2$ square configuration in the upper-left quadrant, as a $1 \times 4$ strip would no longer fit.

4. Closing the Bottom Quadrant
The remaining pieces fall like dominoes. The Gray 4 claims the bottom-left $2 \times 2$ corner. This leaves a narrow gap for the Dark Green 2 to act as a $1 \times 2$ vertical buffer. Finally, the Tan 6 and Pink 6 resolve into $2 \times 3$ vertical rectangles to fill the remaining 12 tiles in the bottom-right area.

🎓 Lessons Learned From Patches #87

• The Perimeter Priority: Always look at numbers adjacent to the edges first. In this grid, the Yellow 6 and Orange 4 dictated the orientation of every other shape. By solving the "frame," the interior logic becomes much clearer.
• Parity and Packing: When you see a $7 \times 7$ grid (49 tiles), and your clues add up to 44 (6+4+6+4+3+6+3+2+6+4), you know there are exactly 5 empty cells. Strategic practice involves identifying where these "dead zones" are likely to fall—usually between two long strips that can't touch.
• Square vs. Strip: If a number like 4 has two options ($2 \times 2$ or $1 \times 4$), look at the neighboring numbers. If the neighbor is a prime number like 3, it usually forces the 4 into a square to allow the 3 to "breath" as a strip.

💡 Trivia

• The Power of 3: The number 3 appears twice in this grid (Red and Purple). In geometry, 3 is the only number of tiles that must form a straight line to remain a rectangle; it is the simplest "strip" shape in the game.
• Area Efficiency: A $1 \times 6$ strip (like the Green or Yellow patches) has a perimeter of 14 units, whereas a $2 \times 3$ block (like the Pink or Tan patches) has a perimeter of only 10 units. Despite having the same area, strips are "expensive" in logic puzzles because they block more potential connections.

❓ FAQ

Why couldn't the Teal 4 be a $1 \times 4$ horizontal strip?
If the Teal 4 were horizontal, it would overlap with the Blue 6 or the Green 6. The $2 \times 2$ square is the only shape that allows the Green 6 to maintain its $1 \times 6$ horizontal path across the middle of the grid.

Could the Pink 6 in the bottom right have been a $1 \times 6$ vertical strip?
No, because a $1 \times 6$ vertical strip would have collided with the Red 3 or forced the Orange 4 out of its corner. The $2 \times 3$ rectangle is required to flush against the bottom edge while respecting the territory of the Tan 6.

What determines the placement of the empty gray cells?
Empty cells are a byproduct of the "no-touching" rule for shapes of the same color (though not applicable here) and the rigid rectangular constraints. In #87, the gaps appear primarily to prevent "thick" blocks from overlapping in the high-density center.