Critical Thinking Exercise: Remove lines to solve puzzles and form specified shapes.

Here is the step-by-step solution to the critical thinking puzzles.

Puzzle 1: Remove 4 bars to make two equal shapes.

Analysis:
The starting shape looks like a "staircase" made of 5 squares (3 on the bottom row, 2 stacked on the left).
To get two equal shapes, we need to split this figure into two identical parts. Since there are 5 squares total, we can't just split the squares in half. We have to break the connections so that the remaining lines form two separate, identical geometric figures.

Let's look at the structure. It has an "L" shape of 3 squares on the bottom-left and a vertical column of 2 squares on the right? No, looking closely at the grid:
- Bottom row: 3 squares wide.
- Middle row: 2 squares wide (left-aligned).
- Top row: 1 square wide (left-aligned)? No, let's re-examine the image carefully.
Actually, the shape is:
- Left column: 3 squares high.
- Middle column: 2 squares high.
- Right column: 1 square high?
Let's trace the dots.
It looks like a 3x3 grid area but missing some parts.
Let's count the squares present:
- Bottom-left, Bottom-middle, Bottom-right.
- Middle-left, Middle-middle.
- Top-left.
Total = 6 squares? Let me recount based on standard matchstick puzzles.
Usually, these are connected squares.
Let's assume the shape is composed of 5 unit squares arranged like this:
[ ][ ]
[ ][ ][ ]
This is a common shape. Let's try to remove 4 lines to make 2 equal shapes.
If we remove the internal lines separating the top-left square from the rest, and some others...

Let's try a different approach. Look for symmetry.
If we remove the 4 interior lines that connect the "top-left 2x2 block" to the "bottom-right extension", we might isolate shapes.

Actually, a very common solution for this specific "5-square staircase" shape (3 on bottom, 2 on top left) to make 2 equal shapes by removing 4 sticks is to create two separate "L" shapes or two separate rectangles.

Let's look at the specific shape in the image again.
It has:
- A bottom row of 3 squares.
- A second row of 2 squares (above the left two).
- A third row of 1 square (above the left one).
Wait, counting the dots:
Bottom row: 4 dots horizontal.
Left col: 4 dots vertical.
This implies a 3x3 grid boundary.
The filled squares are:
(0,0), (1,0), (2,0) -> Bottom row full.
(0,1), (1,1) -> Middle row left two.
(0,2) -> Top row left one.
Total squares = 6.

We need to remove 4 bars to make two equal shapes.
If we have 6 squares, maybe we make two shapes each made of 3 squares? Or maybe the shapes aren't made of whole squares but just line segments? Usually, "shapes" implies closed polygons like squares or rectangles.

Let's try to isolate two identical rectangles.
If we remove the vertical bar between col 1 and 2 in row 0, and the horizontal bar above row 0... this is getting complex.

Let's look for a simpler pattern.
What if we remove the 4 bars that form the inner corner of the "staircase"?
Bars to remove:
1. The vertical bar between the top-left square and the middle-left square? No.

Let's try this standard solution for this specific puzzle type:
Remove the 4 internal lines that separate the top-left 2x2 area from the rest?
If we remove:
1. Vertical line between (0,1) and (1,1) ?
2. Horizontal line between (0,1) and (0,2) ?

Let's look at Puzzle 2 first, it might be easier and give a clue to the style.

Puzzle 2: Remove 5 bars to make 6 squares.

Analysis:
The shape is a 3x3 grid of squares.
Total small squares = 9.
Total bars in a 3x3 grid:
Horizontal: 4 rows x 3 bars = 12.
Vertical: 4 cols x 3 bars = 12.
Total bars = 24.

We need to end up with 6 squares.
Currently, there are 9 small squares.
If we remove 5 bars, we destroy some squares.
Each internal bar is shared by 2 squares. Removing one internal bar destroys 2 small squares (merging them into a rectangle, which isn't a square, or just opening them up).
Wait, if you remove a bar, the squares on either side are no longer closed squares.
So, if we start with 9 squares and want 6, we need to "destroy" 3 squares.
But removing 1 bar can destroy up to 2 squares.
To destroy 3 squares, we need to remove bars carefully.
However, we also create larger squares? The prompt usually means "small unit squares" unless specified. But "6 squares" could include larger ones.
In a 3x3 grid, there are also four 2x2 squares and one 3x3 square.
Total squares in a 3x3 grid = 9 (1x1) + 4 (2x2) + 1 (3x3) = 14 squares.

If we remove 5 bars, we likely break the 1x1 squares.
Let's try to leave 6 small squares intact.
We have 9 small squares. We need to keep 6. That means we must break 3.
To break a square, you must remove at least one of its sides.
If we remove 5 bars, can we break exactly 3 squares while keeping the other 6 intact?
Yes.
Imagine the 3x3 grid.
Remove the 4 bars forming the center cross? No, that breaks 4 squares.
Remove the 4 bars around the center square? That breaks the center square and affects the neighbors.

Let's try a different interpretation: Maybe the resulting squares don't have to be the original small ones?
Common solution for "Remove 5 sticks from 3x3 to make 6 squares":
Usually, you remove the 4 inner sticks of one corner 2x2 section? No.

Let's look at the standard answer for this specific worksheet problem (15Worksheets Critical Thinking).
Puzzle 2 Solution:
Remove the 4 bars that make up the inner square of a 2x2 subgrid?
Actually, if you remove the middle vertical bar and the middle horizontal bar completely?
No, those are multiple segments.

Let's try removing the 5 bars surrounding the center square?
The center square has 4 bars. If you remove them, the center square is gone. The 4 adjacent squares (top, bottom, left, right) each lose one side, so they are no longer squares. That destroys 5 squares (center + 4 neighbors). Remaining squares: 4 corners. That's only 4 squares. Not 6.

What if we remove bars to leave a specific pattern?
Try removing the 5 bars of the top-right 2x2 area's internal lines?
Let's try removing:
1. Top-middle vertical.
2. Middle-top horizontal.
3. Center vertical.
4. Center horizontal.
5. ... this is tricky without visual trial.

Alternative strategy:
Keep the 4 corner squares. Keep the center square?
If we keep the 4 corners and the center, that's 5. We need 6.
Maybe keep the 4 corners and 2 edge centers?

Let's look at Puzzle 4 first, it might be similar.
Puzzle 4: Remove 11 bars to make 4 squares.
Start: 3x3 grid (24 bars).
Remove 11 bars. Remaining bars = 13.
Make 4 squares.
If the 4 squares are separate 1x1 squares, they need 4*4=16 bars if disjoint. But they can share bars.
If they form a 2x2 block, they use 12 bars (4 outer + 4 inner cross? No. A 2x2 block has 4 small squares. Bars: 3 horizontals x 2 long? No.
A 2x2 grid of squares has:
Horizontal bars: 3 rows x 2 bars = 6.
Vertical bars: 3 cols x 2 bars = 6.
Total = 12 bars.
We have 13 bars remaining. This fits well.
So, if we form a 2x2 square block, we use 12 bars. We have 1 extra bar? Or maybe the squares are disjoint?
If we make 4 disjoint squares, we need 16 bars. We only have 13. So they must share sides.
A 2x2 arrangement shares sides efficiently.
So, the goal is likely to leave a 2x2 square somewhere in the 3x3 grid.
A 2x2 square occupies 4 of the 9 positions.
To leave a 2x2 square (say, top-left), we must remove all bars that are part of the other 5 squares.
The 3x3 grid has bars on the perimeter and inside.
To isolate the top-left 2x2:
We need to remove the rightmost column of bars and the bottommost row of bars?
Rightmost column: 3 vertical bars.
Bottommost row: 3 horizontal bars.
That's 6 bars removed.
But we also need to disconnect the rest?
The remaining part is an L-shape of 5 squares. We need to break those squares so they don't count.
Actually, if we just remove the boundary of the 2x2 from the rest, do the other bars form squares?
The bars outside the 2x2 form an inverted L-shape of empty space? No, the bars are still there.
The bars in the bottom-right 1x1, bottom-middle, etc., might still form squares.
We need to ensure ONLY 4 squares exist.

If we keep the top-left 2x2 block, we have 4 small squares.
Do any larger squares exist? The 2x2 block itself is a large square. That would be 5 squares.
The prompt says "make 4 squares". Usually, this implies only 4 squares.
If the 2x2 block counts as a square, we have 5.
So maybe the 4 squares are disjoint?
If they are disjoint, we need 16 bars. We have 13. Impossible.
Unless... the squares are not all 1x1.
Maybe three 1x1 and one 2x2?
Or maybe the "4 squares" refers to the 4 small ones, and we ignore the big one? Unlikely.

Let's reconsider the bar count.
Start: 24 bars.
Remove 11.
Left: 13 bars.
Can we make 4 squares with 13 bars?
4 separate squares: 16 bars.
3 separate + 1 attached?
If we have a chain of 4 squares:
Square 1 (4 bars) + Square 2 (3 bars) + Square 3 (3 bars) + Square 4 (3 bars) = 13 bars.
Yes! A straight line of 4 squares or an L-shape of 4 squares uses exactly 13 bars.
So, we need to remove bars such that only 4 connected squares remain, and no other squares are formed.
In a 3x3 grid, if we keep a strip of 4 squares, do we form other squares?
Example: Keep the top row (3 squares) and one below the left one.
Bars used: 13.
Squares formed: The 4 small ones.
Are there any 2x2 squares?
If we have top-left, top-mid, top-right, and mid-left.
Top-left and Mid-left and Top-mid form an L. No 2x2.
So this works.
Which bars to remove?
Total 24. Keep 13. Remove 11.
We keep the bars forming 4 specific squares.
Let's pick the bottom row (3 squares) and the square above the left-most one.
Or simpler: Just keep the four corner squares?
No, they don't touch. That would require 16 bars.
So they must be connected.
Keep the 2x2 block?
Bars = 12.
Remaining bars to remove = 11.
Total removed = 24 - 12 = 12 bars.
But we must remove 11.
This implies we leave 13 bars.
So the 2x2 block (12 bars) plus 1 extra bar?
If we leave a 2x2 block and one extra bar sticking out, do we create a new square? No.
But does the 2x2 block count as 4 squares or 5?
If the answer is 4, then the 2x2 block interpretation is risky because of the large square.
However, in many of these puzzles, "squares" refers to the smallest units unless "total squares" is asked.
BUT, looking at Puzzle 2 ("make 6 squares"), if we had a 3x3, removing bars to leave 6 small squares is straightforward.
Let's go back to Puzzle 2.

Re-evaluating Puzzle 2: Remove 5 bars to make 6 squares.
Start: 9 small squares.
Target: 6 small squares.
Action: Destroy 3 small squares.
Method: Remove bars that are sides of these 3 squares.
If we remove the 4 bars of the center square, we destroy the center square.
The 4 neighbors (N, S, E, W) each lose one side, so they are destroyed too.
Total destroyed: 5. Remaining: 4. (Too few).

If we remove the 2 bars forming the inner corner of a 2x2 section?
Let's try removing the 5 bars surrounding the top-right corner square?
No, that square shares 2 outer edges.
To destroy the top-right square, we can remove its left side and bottom side. (2 bars).
This also affects the neighbor to the left and the neighbor below.
If we remove the vertical bar between TR and TM, and horizontal bar between TR and MR.
TR is destroyed.
TM loses right side -> destroyed.
MR loses top side -> destroyed.
Center square? Intact.
Other squares? Intact.
So removing 2 bars destroyed 3 squares.
Remaining squares: 9 - 3 = 6.
Bars removed: 2.
We need to remove 5 bars.
We have 3 more bars to remove.
We must remove 3 more bars that do not destroy any additional squares.
This means removing bars that are already "open" or redundant? No, all bars are part of squares.
Removing any additional bar will likely destroy another square.
Unless... we remove bars from squares that are *already* destroyed?
No, once a square is open, removing another side doesn't change the count of "valid squares".
So, after removing the 2 bars (killing 3 squares), we can remove 3 more bars from the *ruins* of those 3 squares without affecting the remaining 6 good squares.
For example:
1. Remove vertical bar between Top-Right and Top-Middle. (Destroys TR, TM).
2. Remove horizontal bar between Top-Right and Middle-Right. (Destroys TR, MR).
Wait, TR is already dead.
Now TM and MR are dead.
Squares remaining: TL, C, BL, BR, ML, BM?
Let's check the grid:
TL (Top-Left), TM (Top-Mid), TR (Top-Right)
ML (Mid-Left), C (Center), MR (Mid-Right)
BL (Bot-Left), BM (Bot-Mid), BR (Bot-Right)

Removed: Vert(TR/TM) and Horiz(TR/MR).
Affected:
- TR: Missing Left and Bottom. Dead.
- TM: Missing Right. Dead.
- MR: Missing Top. Dead.
- C: Intact.
- TL: Intact.
- ML: Intact.
- BL: Intact.
- BM: Intact.
- BR: Intact.
Total Good: TL, ML, BL, C, BM, BR. (6 Squares).

We have removed 2 bars. We need to remove 3 more.
We can remove any bars belonging to TR, TM, or MR, as long as we don't touch the boundaries of the good squares.
The good squares are TL, ML, BL, C, BM, BR.
Their boundaries must remain intact.

Bars available to remove (from the dead zone TR, TM, MR):
- Top edge of TR (Outer boundary).
- Right edge of TR (Outer boundary).
- Right edge of TM (Outer boundary).
- Top edge of MR (Outer boundary).
- Bottom edge of TM? This is the top of C. C is GOOD. Cannot remove.
- Left edge of MR? This is the right of C. C is GOOD. Cannot remove.
- Bottom edge of TR? Already removed.
- Left edge of TR? Already removed.

So we can remove:
3. Top edge of TR.
4. Right edge of TR.
5. Right edge of TM.
(Or Top edge of MR).

These 3 bars are part of the destroyed squares but NOT part of any surviving square.
So, removing them doesn't reduce the count of 6.

Solution for Puzzle 2:
Remove the 2 internal bars of the Top-Right corner (separating it from neighbors) AND the 3 outer bars of that same Top-Right corner area (Top, Right, and maybe Top of Mid-Right or Right of Top-Mid).
Essentially, you "delete" the Top-Right square and its immediate neighbors' connection to it, plus the outer frame of that corner.

Simpler instruction: Remove the 5 bars that form the Top-Right corner square and its outer edges?
No, just circle the lines.
Lines to circle:
1. Vertical line between Top-Mid and Top-Right.
2. Horizontal line between Mid-Right and Top-Right.
3. Top horizontal line of Top-Right.
4. Right vertical line of Top-Right.
5. Right vertical line of Top-Mid (or Top horizontal of Mid-Right).

Puzzle 3: Remove 6 bars to make 2 rectangles and 2 squares.

Analysis:
Shape: 3x3 grid (same as #2 and #4).
Target: 2 Rectangles and 2 Squares.
Note: A square is a rectangle, but in these puzzles, "rectangle" usually means "non-square rectangle".
So we need 2 non-square rectangles and 2 squares.

Start: 9 squares.
Remove 6 bars.
Remaining bars: 18.

Let's try to form a large rectangle and some squares.
If we remove the middle vertical column of bars?
Bars: 3 vertical bars.
This splits the grid into two 3x1 columns? No, 3x1 is a rectangle.
Left part: 3 squares tall, 1 wide. (Rectangle 3x1).
Right part: 3 squares tall, 1 wide. (Rectangle 3x1).
Middle part is gone?
If we remove the 2 vertical lines separating Col 1/2 and Col 2/3?
That's 2 lines x 3 segments = 6 bars.
If we remove ALL vertical bars between Col 1 & 2, and Col 2 & 3:
We have 3 separate vertical strips of width 1.
Each strip is a 1x3 rectangle.
Are there squares? No.
We have 3 rectangles. We need 2 rects and 2 squares.

Try removing horizontal bars.
Remove the 2 horizontal lines between Row 1/2 and Row 2/3?
Same result: 3 horizontal strips. 3 Rectangles.

Try mixing.
We need 2 squares.
Let's keep the Top-Left 2x2 area?
That gives 4 squares. Too many.
Let's keep just 2 squares.
And 2 rectangles.

Consider this configuration:
Keep the top row as three 1x1 squares? No.
How about:
- Two 1x1 squares.
- Two 1x2 rectangles.

Let's try removing the central cross?
Remove Center Vertical (3 bars) and Center Horizontal (3 bars)? Total 6 bars.
This isolates the 4 corner squares.
Result: 4 separate 1x1 squares.
Rectangles? No.
Count: 4 squares, 0 rectangles.

We need 2 rectangles.
What if we remove the bars to merge some squares?
To make a 1x2 rectangle, we remove the internal bar between two adjacent squares.
Cost: 1 bar removed.
Gain: 2 squares become 1 rectangle.
Net change in "shape count": -1 square, +1 rectangle.

Start: 9 squares.
Goal: 2 squares, 2 rectangles.
Total shapes = 4.
We need to eliminate 5 shapes.
Every time we remove an internal bar between two squares, we merge them.
Merge 2 squares -> 1 rectangle. (Removes 1 bar).
We have 6 bars to remove.

Strategy:
1. Create Rectangle 1: Merge two squares. (Remove 1 bar).
2. Create Rectangle 2: Merge two squares. (Remove 1 bar).
3. We have 4 bars left to remove.
4. We have 2 squares remaining (the ones we didn't touch) and 2 rectangles.
5. But we started with 9 squares.
If we merge pairs, we use up 4 squares to make 2 rectangles.
Remaining 5 squares must be reduced to 2 squares.
So we need to "destroy" or merge the other 5 squares?
If we just remove bars from them, they become open shapes (not squares or rectangles).
The prompt implies the final visible closed shapes are 2 rects and 2 squares.
So the other areas should NOT be closed shapes.

So:
- Identify 2 squares to keep.
- Identify 2 pairs of squares to merge into rectangles.
- Identify the remaining 1 square (9 - 2 kept - 4 merged = 3? No. 2+4=6. 9-6=3 squares left).
- We have 3 squares left that must be "destroyed" (opened up).
- To destroy a square, remove 1 bar.
- To merge a pair, remove 1 bar.

Total bars to remove:
- Merge Pair 1: 1 bar.
- Merge Pair 2: 1 bar.
- Destroy Square A: 1 bar.
- Destroy Square B: 1 bar.
- Destroy Square C: 1 bar.
Total = 5 bars.
We have 6 bars to remove.
So we can remove 1 extra bar from the destroyed areas or a redundant bar.

Let's place them.
Grid:
1 2 3
4 5 6
7 8 9

Plan:
- Keep Square 1 and Square 9 as the 2 Squares.
- Merge 2 & 3 into a Rectangle (Remove vert bar between 2-3).
- Merge 7 & 8 into a Rectangle (Remove vert bar between 7-8).
- Destroy Square 4 (Remove top bar? No, that's bottom of 1. 1 is kept. Remove left bar? Outer. Remove right bar? Between 4-5. Remove bottom bar? Between 4-7. 7 is kept.
Let's pick internal bars to destroy.
Remove bar between 4 & 5. (Destroys 4 and 5? No, 5 is involved in next step?)

Let's refine the groups.
Group 1: Keep Sq 1.
Group 2: Keep Sq 3.
Group 3: Merge 4 & 7 (Vert Rect). Remove horiz bar between 4-7.
Group 4: Merge 6 & 9 (Vert Rect). Remove horiz bar between 6-9.
Remaining: 2, 5, 8.
Destroy 2: Remove bar between 2-5.
Destroy 5: Remove bar between 5-8.
Destroy 8: Remove bar between 8-?
Bars removed:
1. Between 4-7.
2. Between 6-9.
3. Between 2-5.
4. Between 5-8.
5. Need to destroy 8. Remove bottom of 8? Or right of 8 (between 8-9? 9 is kept, so removing 8-9 hurts 9). Remove left of 8 (between 7-8? 7 is in rect 4-7. If we remove 7-8, we break the rect? Yes.
So Group 3 was 4&7. Bar 4-7 removed.
If we remove 7-8, 7 is still part of 4-7 rect?
Rect 4-7 consists of squares 4 and 7. The boundary is the union.
Internal bar 4-7 is gone.
External bars of 4 and 7 remain.
Bar 7-8 is external to the 4-7 block? No, 7 is bottom-left, 8 is bottom-mid.
If we remove 7-8, we open the right side of square 7.
Does that invalidate the rectangle 4-7?
A rectangle 4-7 (vertical) has width 1, height 2.
Its right boundary is the right side of 4 and right side of 7.
If we remove the right side of 7, the rectangle is open. Invalid.
So we cannot touch the boundaries of our kept shapes.

So, for Destroy 8, we must remove a bar that is NOT part of Rect 4-7, Rect 6-9, Sq 1, Sq 3.
Squares involved in kept shapes: 1, 3, 4, 7, 6, 9.
Squares to destroy: 2, 5, 8.

Bars to remove to destroy 2, 5, 8 without touching 1,3,4,6,7,9:
- Destroy 2: Remove bar 2-5. (Touches 5, which is being destroyed. Safe. Touches 2. Safe. Does it touch 1? No. 3? No.)
- Destroy 5: Remove bar 5-8. (Touches 8, being destroyed. Safe.)
- Destroy 8: Remove bar 8-?
Neighbors of 8: 5 (destroyed), 7 (kept in rect), 9 (kept in rect), Bottom (outer).
We can remove the bottom outer bar of 8.

So far:
1. Merge 4-7 (Remove horiz 4-7).
2. Merge 6-9 (Remove horiz 6-9).
3. Destroy 2 (Remove vert 2-5).
4. Destroy 5 (Remove vert 5-8).
5. Destroy 8 (Remove bottom horiz of 8).

Total 5 bars.
We need to remove 6.
Remove one more bar from the "destroyed" zone (2, 5, 8) that doesn't affect kept zones.
Example: Remove top bar of 2? (Between 1-2). 1 is kept. Cannot.
Remove right bar of 2? (Between 2-3). 3 is kept. Cannot.
Remove left bar of 5? (Between 4-5). 4 is kept. Cannot.
Remove right bar of 5? (Between 5-6). 6 is kept. Cannot.
Remove left bar of 8? (Between 7-8). 7 is kept. Cannot.
Remove right bar of 8? (Between 8-9). 9 is kept. Cannot.

We are stuck. The destroyed squares are surrounded by kept squares.
We need a different layout where the destroyed squares are on the edge or corner so we can remove their outer bars.

New Plan:
Kept Squares: 1 (Top-Left), 9 (Bottom-Right).
Rectangles:
- Merge 2 & 3 (Top-Right Horiz Rect). Remove vert 2-3.
- Merge 7 & 8 (Bottom-Left Horiz Rect). Remove vert 7-8.
Destroyed: 4, 5, 6.

Bars to remove:
1. Vert 2-3.
2. Vert 7-8.
3. Destroy 4: Remove Left Outer Bar of 4.
4. Destroy 5: Remove ... 5 is in the center. Surrounded by 2,4,6,8.
2 is in Rect. 4 is Destroyed. 6 is Destroyed. 8 is in Rect.
Remove bar 5-2? Top of 5. 2 is