LinkedIn Patches #72 Answer

🧩 Deep Logic Analysis

This grid was a fantastic exercise in using the grid's boundaries to create certainty. The solution unfolds not from the middle, but from the absolute edges.

1. The Unbreakable Starting Point: The most critical clue is the Gold 6 on the far-right edge. The grid is 6 cells tall. The only possible shapes for a 6 are 1x6, 6x1, 2x3, or 3x2. Since the clue is on the edge of a 6-tall grid, the only configuration that fits perfectly without leaving impossible gaps is a 1x6 vertical strip. This locks in the entire rightmost column immediately, providing a hard boundary for all other shapes.

2. The Factor-Forced Rectangle: Next, we turn to the Blue 12. On a 6x6 grid, a 12-cell shape has limited possibilities. Factors of 12 are (1,12), (2,6), and (3,4). Both 1x12 and 2x6 (horizontally) are too large for the grid. A 2x6 (vertically) would occupy two full columns, but its starting clue placement prevents this. This forces the Blue 12 to be a 3x4 or 4x3 rectangle. Given its central position at the top, a 3-wide by 4-tall rectangle is the only fit that doesn't immediately crash into other clues.

3. The Domino Effect: With the Gold 1x6 and Blue 3x4 in place, the rest of the puzzle falls like dominoes. The Purple 6 clue sits at the bottom, directly under the newly placed Blue rectangle. The space available is exactly 3 cells wide and 2 cells tall. This perfectly accommodates a 3x2 rectangle for the Purple 6.

4. Solving by Containment: The final two shapes, Red and Green, had no numbers. We solve them by seeing what's left.

   • In the bottom-left, the Orange, Blue, and Purple shapes have created a perfect 2x2 corner. This must be the Red shape, revealing its area is 4.
   • On the left, the Orange 4 can now only be a 1x4 vertical strip to fill the remaining space without creating an invalid 2x1 gap at the bottom.
   • This leaves a 1x4 vertical space for the Green shape, defining its area as 4.

The key to this puzzle was starting from the outside and working in. With consistent practice, recognizing these edge-based locks becomes second nature.

🎓 Lessons Learned From Patches #72

1. The Full-Column Lock: When a clue's area (like the Gold 6) has a factor that matches the grid's dimension (a 6-cell grid), and the clue is on that edge, it's almost always a full-length strip. This is one of the most powerful starting moves in any Patches puzzle.
2. Factor Forcing on a Finite Grid: Always check the factors of large numbers against the grid's dimensions. The Blue 12 was immediately constrained from a 2x6 or 1x12 to a 3x4 shape simply because the other configurations wouldn't physically fit. This drastically reduces the number of possibilities you need to consider.
3. Solve the Knowns to Reveal the Unknowns: You don't always need a number to solve a shape. By placing the Gold, Blue, and Purple rectangles with certainty, we perfectly outlined the spaces for the Red and Green shapes. Their area and dimensions were revealed by the other pieces, not by their own clues.

💡 Trivia

• The number 12 is a "highly composite number," meaning it has more divisors (1, 2, 3, 4, 6, 12) than any smaller positive integer. This property is why it has been historically fundamental to timekeeping (12 hours, 12 months) and measurement systems (12 inches in a foot).
• The number 6 is the first "perfect number." A perfect number is a positive integer that is equal to the sum of its proper divisors (the divisors excluding the number itself). The proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. The next perfect number is 28.

❓ FAQ

Why couldn't the Gold 6 be a 2x3 rectangle?
A 2x3 rectangle is two cells wide. Placing a two-cell-wide shape against the absolute edge of the grid would require another shape to fill the remaining single column to its left. This is impossible, as no shape can be only one cell wide and also wrap around another piece. Therefore, the only logical fit for a clue on a 6-tall edge was the 1x6 strip that filled the entire column.

How was the area of the Red shape determined without a number?
The Red shape was solved entirely by elimination. Once the certain pieces—the Gold 1x6 strip on the right, the Orange 1x4 strip on the left, and the Purple 3x2 rectangle on the bottom—were placed, they formed a perfect container. The only remaining empty space in that bottom-left quadrant was a 2x2 square. The Red shape had no choice but to fill that space, defining its area as 4.

Why was the Blue 12 forced into a 3x4 shape instead of a 2x6?
A 2x6 rectangle can only fit vertically on a 6x6 grid. The clue for the Blue 12 was positioned centrally in the top row. Placing a 2x6 strip there would have occupied two full columns from top to bottom. This would have conflicted with the space needed for the Purple 6 clue located at the bottom of the grid. The 3x4 configuration was the only one that fit in the top section without violating the space required for other clues.