THE SHAPES BETWEEN THE STICKS
Nine squares.
More than meets the eye.
A brain snack, with room to think.
THE ANSWER OPENS ANOTHER DOOR
You have an answer. Explore how it works, then take the idea somewhere new.
See why it works ↓THE IDEA BEHIND THE ANSWER
Choose two of the four vertical grid lines and two of the four horizontal lines. Each pair can be chosen in six ways: 6 × 6 = 36 original rectangles. Your seven gaps have to interrupt every one.
A broken little square can sit inside a larger rectangle that is still complete. The boundary is what matters.
ONE MORE AHA
Imagine the nine squares as rooms and the sticks as walls. This gives us a way to reason about every possible arrangement, without trying them all.
Removing a wall can join two rooms, or open a room to the outside. Each removal reduces the number of enclosed rooms by at most one.
Starting with nine, six gaps leave at least 9 − 6 = 3 enclosed rooms. Removing a wall inside an already joined region does not help this count.
An enclosed room of one cell is a square. Two joined cells make a rectangle. So each remaining room needs at least three cells to have any chance of an irregular outline.
The outermost rectangle must also be broken, opening at least one cell to the outside. That leaves at most eight cells for enclosed rooms.
That contradiction rules out six removals—and fewer. A valid seven-gap arrangement proves that seven is the minimum.
WHAT YOUR SEVEN GAPS MAKE POSSIBLE
Matching letters belong to one room. ↗ marks a cell connected to the outside. Colours group the cells; the stick diagram above shows the actual walls.
The idea to take away
Try changing what you count. Sticks are hard to track; rooms make a useful limit visible. A good representation can turn a search into an explanation.