Patches #53 Hints & Answer

Patches #53 is solved by covering the whole board with rectangles, one clue per block. We split it into 8 steps below: fix the constrained clues first, and the flexible any-shape clues fall into place last.

Step-by-step solution

1. Step 1

   Work the square clue around row 1, col 6. With 1 cells to cover and a fixed shape, only one rectangle fits without overlapping a neighbour. Commit to it and the board tightens.

2. Step 2

   Look at the square clue near row 3, col 4. Its number 1 fixes the rectangle’s area, so draw the block that covers exactly 1 cells around it without touching another clue. Lock it in before moving on.

3. Step 3

   The tall clue at row 3, col 6 can only be filled one way: a rectangle of area 5 that matches its proportions. Place it now — once it is down, the cells beside it stop being ambiguous.

4. Step 4

   Work the tall clue around row 4, col 5. With 5 cells to cover and a fixed shape, only one rectangle fits without overlapping a neighbour. Commit to it and the board tightens.

5. Step 5

   Look at the wide clue near row 6, col 1. Its number 5 fixes the rectangle’s area, so draw the block that covers exactly 5 cells around it without touching another clue. Lock it in before moving on.

6. Step 6

   The wide clue at row 1, col 1 can only be filled one way: a rectangle of area 8 that matches its proportions. Place it now — once it is down, the cells beside it stop being ambiguous.

7. Step 7

   Work the square clue around row 3, col 3. With 9 cells to cover and a fixed shape, only one rectangle fits without overlapping a neighbour. Commit to it and the board tightens.

8. Step 8

   Finally resolve the dashed any-shape clue near row 4, col 4. Its rectangle can be any proportion, so fill it last — the surrounding blocks now leave only one valid placement that covers every remaining cell.