Minimize non-touching squares in irregular grid regions
Leave space between every square. Make the arrangement complete. Then ask the harder question: could fewer squares do the job?
No shared edge.
Not even a shared corner.
“Fewest” needs a finish line.
We need to know both what is allowed and when the arrangement is finished. Otherwise, why not stop after one square?
A missing rule in the book: the printed question asks for the fewest non-touching squares, but does not say what makes an arrangement complete. Its answers show several squares with blank spaces between them.
Our workshop adds this finish rule: keep going until no more square can be added without touching an existing square or leaving the region. Then try to finish with as few squares as possible. Both printed arrangements pass this rule, but the book does not explicitly state it.
Each square has a whole-number side length of at least 1. Every cell it covers must belong to the region. Touching the region’s outer boundary is allowed.
No overlap, no shared edge, and no shared corner. We treat a square as a filled piece, like the shaded original answers; nesting one inside another is not allowed.
Even a 1 × 1 square must be unable to fit. Blank cells may remain, but each must touch an existing square.
Count square pieces, not shaded cells. A finished arrangement is not automatically the best one.
Does this pair obey the rule?
Choose a case and look closely at the boundaries.
Two squares meet at just one corner. Are they allowed?
Why is testing 1 × 1 squares enough to know when we are finished?
If any larger square could fit, one of its 1 × 1 cells could also be placed as a square without touching another piece. So if no 1 × 1 square fits anywhere, no larger square can fit either.
An empty cell that merely touches a placed square is unavailable too. Its own boundary would share a point with that square. This is why the checker tests corners as well as edges.
Start big. Leave useful gaps.
The book’s small-region answer uses three squares. Let’s rebuild it and check why its blank cells do not leave room for another square.
Leave no room for one more.
Build on the small region, then try the larger Practice 6 region. Different valid arrangements are welcome; you do not have to copy the worked example.
Rows count downward; columns count to the right. Numbers start at 1. The preview uses the chosen cell’s top-left grid corner.
Your squares
0 piecesFinding room for one more is a check, not a claim that adding a tiny square is the best strategy.
original construction · Reveal when you are ready
The book’s three-square construction
What the checker verifies: the pieces fit, are separate, and leave no legal extra square under our finish rule.
This changes the current board. Undo brings your previous arrangement back. Checking a guided solution can earn a checkpoint too.
Workshop: draw an irregular region on grid paper. Place grid-aligned squares without allowing them to overlap or touch. Stop only when no additional square can fit. Try again with fewer squares, explaining your changes.
Finished does not mean fewest.
This four-square arrangement leaves no room for another square. We can still improve it by removing a piece and shifting another one.
Compare two three-square answers: must they cover the same area?
No. Our goal is the number of square pieces, not the number of shaded cells. These different three-square arrangements both leave no place for another square under the workshop rule.
The printed construction
Our adjusted construction
How do we know three is the minimum?
One working three-square layout shows “three is enough.” To prove “three is the fewest,” we also have to rule out every smaller completed arrangement.
Test all possibilities with fewer than three.
The page first lists every grid-aligned square that fits inside the small region, including every side length from 1 to 5. It then checks the empty arrangement, each single square, and every unordered pair. Touching pairs are rejected.
| Square count | Sets tested | Separate sets | Finished sets |
|---|---|---|---|
| 0 | — | — | — |
| 1 | — | — | — |
| 2 | — | — | — |
This proof is for our explicitly stated finish rule. It is a teaching addition, not a proof printed in the book.
Why one failed pair is not the proof
These two squares do not touch, but the dots show room for another square. That rejects this pair only.
The audit checks every pair—including pairs of different sizes and pairs placed elsewhere in the region.
What is—and is not—being proved for the large region?
The worked example reference answer reports 8 and shows an eight-square arrangement. The workshop reconstructs that layout and verifies that it also passes our added finish rule.
The small-board audit above does not search the large region. A “finished” message on the large board verifies your construction; it does not itself prove that seven squares are impossible. The number 8 is presented as the worked example’s reported answer, not as a minimum established by this in-page audit.
What makes an arrangement convincing?
Check the rules, the finish condition, and the strength of a minimum claim. You can revise your answers and try again.
Four checkpoints complete.
You have checked separation, verified both original regions under the workshop rule, and explained the difference between a construction and a minimum proof.
My explanation
Learning notes, the missing condition & teaching additions
What the worked example states: draw squares along the grid lines, with no common point between the squares, and ask for the least number. Example 7 recommends considering large squares first and then adjusting, and reports 3. The reference section reports 8 for Practice 6.
What is missing: the printed prompt does not specify what the selected squares must collectively accomplish. Without such a condition, the empty set—or one square, if at least one is required—would make minimization trivial.
Explicit modeling choice:not confirmed original wording. The square is treated as its whole filled area, not only its outline. Squares may touch the region boundary. Any positive integer side length is allowed; 1 × 1 squares are included. Both printed layouts satisfy these conventions.
Exact redrawings: the small region has 46 unit cells within a 7-column, 8-row bounding rectangle. The large region has 240 unit cells within a 17-column, 16-row bounding rectangle. The large worked solution uses sides 5, 3, 5, 3, 7, 2, 2, 4.
Checker: two closed, axis-aligned square pieces conflict unless one is strictly to the left, right, above, or below the other. Equality at a boundary counts as touching. A target-count checkpoint verifies a finished construction using no more than the worked example’s reported count; it does not distinguish independent work from guided work.
Teaching additions: contact demonstrations, the completion convention, staged construction, unavailable-cell overlays, coordinate editing, hints, the four-to-three adjustment, area comparison, exhaustive small-board audit, and exit ticket. The audit generates all 113 fitting small-board squares and all 6,328 unordered pairs; it does not assume that the worked example construction is unique. Its minimum conclusion is restricted to the small region and the stated workshop model. No large-region minimum proof is claimed by the page.
Using the page: every diagram, style, and calculation is included in this HTML file. No internet connection is required. Optional browser storage saves both boards, checkpoints, and your explanation locally. Blocked storage does not prevent play. Moving or renaming the file may change its storage context. Printing produces a reading-and-reflection version, not a replacement interactive board.