Bonus puzzle studio · spatial constraints

Pack T-Shaped Tanks from Row-and-Column Clues

Rotate seven-cell tanks, keep every pair completely separate, and use each numbered cell as a row-and-column counting clue. Start with the most restrictive clue, place one tank, and update every affected line.

Grade 510 missionsCore works offlineRotation + line countsExact audits

Your progress and constructions are saved in this browser when storage is available.

Your tasks: pack two tanks, then three

A tank is a seven-cell shape with a body and a head. You choose each tank’s position and direction.

Two-tank puzzle: place exactly two tanks so the three clues 2, 5, and 4 all match. Missions 1–2 teach the shape and counting rules. Missions 3–4 inspect and reconstruct one worked packing; Mission 5 is your construction.
Three-tank puzzle: use the same rules to place exactly three tanks and satisfy all five clues. Missions 6–7 study the restrictive clues and a worked packing; Mission 8 is your construction.

A finished packing must pass every rule: keep all tanks inside the board, leave clue cells empty, match every clue, and allow no overlap, edge contact, or corner contact between tanks. Matching just one clue is not enough.

The tank

A 3-by-2 body plus one head

Every tank occupies exactly seven unit cells: six cells in a rectangular body and one head attached to the middle of a long side. A tank may face up, right, down, or left.

1 head + (3 × 2) body = 7 cells

How a clue works

Count the occupied cells on two crossing lines

clue value = tank cells in its row or its column

Count occupied cells, not the number of tanks. A tank cell is counted once if it lies on either highlighted line. Since the crossing clue cell stays empty, you can add the row count and column count directly. In this page’s original-based working convention, clue cells are unavailable for tanks.

The printed diagrams leave clue cells uncovered and say that two tanks have no common point. The checker therefore forbids covering clue cells and forbids tanks from touching at an edge or corner. These conventions are stated here rather than hidden inside the app.

Read every condition

Decode the tank and packing rules

Not complete

Seven occupied cellsOne head cell and a 3-by-2 body.

Four rotationsThe head may face up, right, down, or left.

No common pointTwo tanks may not overlap or touch at an edge or corner.

Clue linesEach number counts occupied cells in its whole row and column.

Checkpoint

Body cells

Head cells

Total cells

May two tanks touch at one corner?

Cross-sections

Rotate a tank and scan one row or column

Not complete

Choose a view

Scan line

Line number inside the 3×3 box

Facingup

Occupied cells on line1

Head on line?yes

Total tank cells7

Depending on where a row or column cuts the tank, one tank can contribute 1, 2, or 3 occupied cells to that single line. A numbered clue combines two lines. Its cell is outside the tank: if both lines meet the tank, they meet at one of the two empty bounding-box corners, contributing 1+2=3. Thus one tank also contributes at most 3 to a legal clue.

Smallest positive line contribution

Largest line contribution

A body row across a head-up tank contributes

Worked example 3

Audit the clues 2, 5, and 4

Not complete

Learn to read a supplied two-tank solution. The tanks are already placed here. Select each clue to see how the two tanks contribute to its row and column. These questions concern this displayed example; Mission 5 will ask you to place the tanks yourself.

Select a clue

1 + 1 = 2

Tank	Cells on this row/column	Contribution

Clue 2 is split as

Clue 5 is split as

Clue 4 is split as

Forced orientation

Reconstruct the two original tanks

Not complete

Reconstruct the same two-tank example. Begin with the clue board, then follow the deductions to reveal the tanks. Answer the direction questions about the completed worked packing, even if you have revealed only part of it.

Stage0 / 4

Tanks shown so far0 / 2

Clues matched by tanks shown so far0 / 3

Orientation relation—

In the complete example, the upper-left tank faces

In the complete example, the lower-right tank faces

In the complete example, are the directions perpendicular?

Validate a complete packing, then account for its cells and clues.

Construct and explain the example

Not complete

Your two-tank packing. Choose positions and directions, satisfy all three clues, and keep the tanks from touching. Then explain how you checked the result. The optional search can help you compare complete solutions.

Reason it through

Each tank has 7 cells, so two tanks occupy 14. If a clue needs 5 and one tank contributes 3, the other must contribute 2. Clue cells stay empty, and even corner contact between tanks is forbidden. The search below is optional.

Choose a tank and direction, then select the top-left cell of its 3×3 bounding box, even if that corner of the box is not occupied by the shape. Choosing a new location replaces that tank’s previous placement. The box must stay entirely inside the board.

Placement controls

Placed0 / 2

Clues matched0 / 3

Touch conflicts0

Statusincomplete

Clue	Actual	Target

Added exact audit

The worked example prints one construction. This audit generates every legal placement, every no-touch pair, and every pair satisfying all three clues.

Legal placements—

Raw unordered pairs—

No-touch pairs—

Exact solutions—

Run the audit after exploring the board.

Two tanks occupy how many cells altogether?

A clue needs 5; one tank contributes 3. What must the other contribute?

How many tank cells may cover a printed clue cell?

How many touching tank pairs are allowed?

original Practice 3 strategy

Start with 7 and especially 1

Not complete

A new puzzle with three tanks and five clues. This supplied packing shows a useful way to satisfy them. Study the contributions at clues 7 and 1 before constructing your own packing in Mission 8.

Inspect a clue

3 + 3 + 1 = 7

7 = 3 + 3 + 1This worked packing uses two contributions of 3 and one of 1. The sum 7 could also split as 3 + 2 + 2; only checking the positions and all other clues can show whether a proposed split works.

1 = 1Only one head may appear anywhere in that clue’s row or column.

5 can split several waysUse it after the rare clues have fixed orientations and positions.

In the displayed worked packing, the three contributions to clue 7 are

Why is clue 1 so restrictive?

Total tank cells for three tanks

Reference construction

Reconstruct the three-tank practice arrangement

Not complete

Reconstruct the three-tank worked example. Reveal the tanks one at a time. The counters describe the portion shown so far; the direction and clue questions below refer to the completed packing.

Stage0 / 4

Tanks shown so far0 / 3

Occupied cells shown so far0 / 21

Clues matched by tanks shown so far0 / 5

Tank	Facing	Bounding-box top-left

In the complete example, the left tank faces

In the complete example, the top tank faces

In the complete example, the right tank faces

In the completed worked packing, how many clues match?

Any complete packing satisfying every stated condition is accepted.

Construct and explain the practice packing

Not complete

Your three-tank packing. Place all three tanks, satisfy all five clues, and check every pair for touching. You may find a different arrangement from the worked example; the checker accepts any valid packing.

Reason it through

For clue 7, contributions of 3 and 3 leave 1. Once clue 1 has its single head cell, every other tank cell must avoid both of its lines. Validate your board and explain these checks; the complete search is optional.

Tank slots are labeled A, B, C and colored to help you keep track; the tanks are mathematically identical. Exchanging labels on the same occupied shapes is not a new packing.

Placement controls

Placed0 / 3

Clues matched0 / 5

Touch conflicts0

Statusincomplete

Clue	Actual	Target

Added exact audit and solution gallery

Legal placements—

Raw triples—

No-touch triples—

Exact solutions—

Run the audit to compare all complete packings.

Three tanks occupy how many cells altogether?

Clue 7 already has contributions 3 and 3. What remains?

Clue 1 is already filled by one head. How many more cells may enter its row or column?

How many of the five practice clues must match?

Independent review

Tank-packing workshop

Not complete

1. How many cells does one tank occupy?

2. How many cells are in its rectangular body?

3. May two tanks touch at a corner?

4. Which split explains the worked example clue 2?

5. Which split does the worked example use first for clue 7?

6. Clue 4 already has 2 cells. How many more are needed?

7. Clue 1 already has one head cell. How many further cells may be added to those lines?

8. A clue cell is counted as an occupied tank cell.

Final check

Exit ticket and certificate

Not complete

1. Possible positive contributions from one tank to one line are

2. The two original-example tanks face up and

3. Three tanks occupy how many cells altogether?

4. Which practice clue should be treated as especially restrictive?

5. A complete construction must satisfy

Optional reflection: Which clue would you investigate first on a new board, and what makes it restrictive?

Your tasks: pack two tanks, then three

A 3-by-2 body plus one head

Count the occupied cells on two crossing lines

Decode the tank and packing rules

Checkpoint

Rotate a tank and scan one row or column

Choose a view

Audit the clues 2, 5, and 4

Select a clue

Reconstruct the two original tanks

Construct and explain the example

Reason it through

Placement controls

Added exact audit

Start with 7 and especially 1

Inspect a clue

Reconstruct the three-tank practice arrangement

Construct and explain the practice packing

Reason it through

Placement controls

Added exact audit and solution gallery

Tank-packing workshop

Exit ticket and certificate

Puzzle Solver