I need to figure out how many holes are in the inner area after unfolding this paper. First, let's understand the setup. There's a big square divided into a 3x3 grid of smaller squares by dashed lines. It's folded in some order and direction, and then one corner of the resulting small square is cut off. After unfolding, we need to find how many holes there are.

The problem doesn't specify how it's folded, so I think I need to assume a common folding sequence or figure out what makes sense. Let me visualize this.

The big square has dashed lines, so it's like a tic-tac-toe board. After folding, it becomes a small square, which I assume is one of the smaller squares from the grid.

It says "the resulting small square," so probably after folding, it's reduced to the size of one small square. Then, one corner of that small square is cut off. Then unfolded back to the big square.

But when you fold paper, you're creasing it, and cutting while folded might affect multiple layers. Then unfolding would show multiple cuts or holes.

The question is about holes on the inner area. What is the inner area? Maybe it means the area inside the original big square after unfolding.

Perhaps "inner area" refers to the regions that were folded inward or something. I'm a bit confused.

Maybe "inner area" means the area that was covered or something. Let me read the problem again carefully.

"a square bit of paper is folded along the dashed lines in some order and direction. one of the corners of the resulting small square is cut off. the piece of paper is then unfolded. how many holes are on the inner area of the piece of paper?"

It doesn't define "inner area." That might be ambiguous. Perhaps in the context of paper folding, "inner area" means the region that was folded upon itself or the layers that were stacked.

Maybe "inner area" means the area that was inside the folds, but I'm not sure.

Another thought: when you fold the paper multiple times, some parts are on the inside of the folds, and some on the outside. But when you cut, the cut might affect multiple layers if they are stacked.

Then, when unfolded, the cuts correspond to multiple holes in the paper.

But the problem mentions "holes on the inner area," which might imply that we care about holes in the regions that were internal during folding.

But I need to know how it was folded to know what the inner area is.

The folding is along the dashed lines, which divide the big square into 9 smaller squares. So, to fold it into a small square, it must be folded in a way that layers are stacked.

For example, one common way is to fold it like a brochure or a map to make a smaller version.

Since it's a 3x3 grid, to get down to one small square size, we probably need to fold it multiple times.

For instance, first fold it in half along one row, then in half again along the columns or something.

Let me denote the big square with positions. Let's say the corners are A, B, C, D for simplicity, but since it's a grid, let's label the small squares.

Assume the big square has 9 small squares: let's number them from 1 to 9, row-wise, so (1,1), (1,2), (1,3), (2,1), etc., up to (3,3).

But when folding, we fold along the lines between them.

The dashed lines are the boundaries, so folding along vertical or horizontal lines.

To reduce to a small square, we might fold outer parts over the center or something.

The problem says "in some order and direction," so it doesn't specify, but probably for the cut to be symmetric or something, it might be folded in a standard way.

Perhaps it's folded to the center small square.

Let's think about the sequence.

Suppose we want to fold the 3x3 grid so that all 9 small squares are stacked on top of one another or in layers.

But when folded, it becomes a small square, meaning the size is reduced, so the area is smaller, so multiple layers are stacked so that the footprint is that of one small square.

Then, we cut one corner of this small stacked square.

Then unfold, and each layer that was cut will have a hole when unfolded, but since it's paper, cutting one corner might cut multiple layers if stacked.

Then when unfolded, there might be multiple cutouts.

But the number of holes might depend on how many layers were stacked at the cut point.

Also, "holes on the inner area" – I need to clarify that.

Perhaps "inner area" means the area that was folded underneath or something.

Another idea: in paper folding, when you fold something like this, the "inner area" might refer to the center square or the squares that are folded inward.

But let's look at the image; I can't see it, but the problem mentions "the resulting small square," so after folding, it's a small square, which I think is one of the small squares from the grid.

Let's assume that the folding is such that the center small square is the one that everything is folded onto, or something.

Perhaps it's folded to the bottom-left small square or something; it doesn't specify.

To make it general, I need to think about how many times each part is folded.

Since it's folded along grid lines, the creases are at the boundaries.

Let me think about the folding as a process of halving or something.

For a 3x3 grid, to fold it to one-ninth the area, we need to fold it twice, I think.

For example, first fold vertically, bringing the left and right columns together, but since it's 3 columns, folding vertically might not halve it if odd number.

3 is odd, so folding might not be symmetric.

For instance, we can fold the left column over the middle, or right over middle, etc.

Similarly for rows.

Then after that, we have a 2x3 or 3x2 or something, but let's see.

Suppose we fold along the vertical line between column 1 and 2, and column 2 and 3.

But to reduce size, we might fold the outer columns over the middle.

Let's define: let the columns be 1,2,3 from left to right, rows 1,2,3 from top to bottom.

First, fold column 1 over column 2. So now, column 1 is on top of column 2, and column 3 is still there. But the size: now it's 2 columns wide, but columns 1 and 2 are stacked, so the visual width is one column, but with two layers for the left part, and column 3 separate? No.

When you fold column 1 over column 2, since they are adjacent, after folding, the left part is folded over, so now the paper has columns 2 and 3, but column 1 is on top of column 2, so the left side is column 1 (folded) and column 2, but since folded, the footprint is from the left edge of column 2 to column 3, but column 1 is on top, so the area covered is that of column 2 and 3, with column 1 layered on top of column 2.

I'm confusing myself.

Let me think of the paper as having width and height.

Big square: let each small square be size 1, so big square is 3x3.

Dashed lines at x=1, x=2, y=1, y=2, say, with (0,0) at bottom-left or top-left; let's use coordinates.

Set coordinate system: let the big square go from (0,0) to (3,3), with dashed lines at x=1, x=2, y=1, y=2. So small squares: square (i,j) from x=i to i+1, y=j to j+1, for i=1,2,3? Better to use indices.

Define the grid: the vertical lines at x=0,1,2,3 and horizontal at y=0,1,2,3, with squares between them.

Squares: for i=1,2,3 and j=1,2,3, square (i,j) from x=i-1 to i, y=j-1 to j, or something. Let's set i and j from 1 to 3, with square (i,j) : x from (i-1) to i, y from (j-1) to j, so for i=1,2,3, j=1,2,3.

So for example, square (1,1): x0 to x1, y0 to y1, (1,2): x0 to x1, y1 to y2, etc. I think I have indices mess.

Standard: let the columns be defined by vertical lines at x=0,1,2,3, and rows by horizontal lines at y=0,1,2,3. So the small squares are defined by the regions: for col c from 1 to 3, row r from 1 to 3, square (c,r) has x from c-1 to c, y from r-1 to r? Let's set c and r starting from 1.

To avoid confusion, let's say the paper has min x=0, max x=3, min y=0, max y=3.

Dashed lines at x=1, x=2, y=1, y=2.

So the grid lines are at x=1 and x=2 (vertical), and y=1 and y=2 (horizontal).

Folding along these lines.

To fold to a small square, we need to fold such that the final size is 1x1, so we need to stack multiple layers.

For example, one common way is to fold it in thirds or something.

Since it's 3x3, we can fold it by bringing the outer parts to the center.

But let's assume a specific sequence, as the problem doesn't specify.

Perhaps for the cut to be on a corner, it might be folded to a corner.

The problem says "one of the corners of the resulting small square is cut off." So after folding, it's a small square, and we cut one corner of it.

Then, when unfolded, the cut will be mirrored or replicated based on the folding.

Also, "inner area" might mean the area that was not the outer layer or something.

I recall that in such problems, "inner area" might refer to the region that was folded upon itself, like the center in a map fold.

But let's try to find a standard way.

Another thought: the "resulting small square" is probably the size of one of the original small squares, so after folding, the visible shape is a small square, with multiple layers inside.

Then, cutting one corner: since it's a corner, and the small square might be oriented, but we cut one corner.

When cut, the cut affects all layers that are stacked at that point.

Then when unfolded, each layer that was cut will have a corner cut off, but only if that layer was at the corner position when cut.

Since we cut the corner of the small square, that corner corresponds to a specific location in the folded state.

But in the unfolded state, that location maps to different points depending on the folding.

So I need to know the folding sequence to know the mapping.

Perhaps the folding is such that it is symmetric, and we cut the corner that corresponds to the original corner or something.

The problem doesn't specify which corner to cut, but it might not matter due to symmetry.

Similarly for folding direction.

But to make it simple, let's assume that the small square we fold to is the bottom-left small square, for example, and we cut its bottom-left corner or something.

But the small square after folding might not correspond to an original small square; it's the size, but the content is stacked.

I think I need to think about the layers.

Let me assume a specific folding sequence.

Suppose we first fold vertically along the line x=1.5? No, the lines are at x=1 and x=2, which are between integers, so for example, folding along x=1, which is between column 1 and 2.

Let's define: the vertical lines at x=1 and x=2 separate the columns.

Folding along x=1: for example, fold the left part (x<1) over the middle or right.

But x=1 is the line, so folding along x=1 means folding the region x<1 over x>1 or vice versa.

Similarly for other lines.

To reduce size, we might fold the outer columns inward.

For example, fold column 1 (x from 0 to 1) over column 2 (x from 1 to 2) along the line x=1.

When we fold along x=1, folding the left part over to the right, so column 1 is folded onto column 2, and now the paper is from x=1 to x=3, with column 1 and 2 stacked at the same y, but since we folded along x=1, after folding, the point at x=1 is the fold line, so the material from x=0 to x=1 is now on top of x=1 to x=2, so the combined area has x from 1 to 2 for the stacked layers, but then column 3 is from x=2 to 3, so the paper is not flat; it's like we have a flap.

After folding column 1 over column 2 along x=1, the paper now has the shape: from x=1 to x=2, we have two layers (column 1 and 2), and from x=2 to x=3, we have one layer (column 3). So the width is from x=1 to x=3, which is 2 units, not 1.

Then we need to fold again to reduce.

Similarly for rows.

To get to a 1x1 size, we need to fold multiple times.

For example, after folding column 1 over column 2, we have something that is 2 units wide (x from 1 to 3) and 3 units high.

Then we can fold this vertically again, but it's not symmetric.

Fold along x=2, for example. x=2 is the line between column 2 and 3.

After first fold, column 1 and 2 are stacked at x=1 to 2, column 3 is at x=2 to 3.

So the paper occupies x=1 to 3, with layers: at x=1 to 2, two layers, at x=2 to 3, one layer.

Now, fold along x=2. x=2 is the boundary between the stacked area and column 3.

If we fold along x=2, we can fold the single layer part over the double layer part.

For example, fold column 3 over the stacked columns along x=2.

Since column 3 is from x=2 to 3, folding along x=2 means folding it to the left, so it goes over the area at x=1 to 2.

After folding, column 3 is now on top of the stacked columns at x=1 to 2, so now at x=1 to 2, we have three layers: column 1,2, and 3, all stacked.

And the paper is now from x=1 to x=2, size 1 unit wide, but 3 units high.

Similarly, we need to fold the rows.

So now we have a tower that is 1x3, so we fold the rows.

For example, fold along y=1 or y=2.

Say, fold the bottom row up or something.

Assume we fold along y=1. y=1 is between row 1 and 2.

First, let's define rows: row 1 bottom, row 2 middle, row 3 top, or vice versa; it doesn't matter, but let's say y from 0 to 3, rows: row 1: y=0 to 1, row 2: y=1 to 2, row 3: y=2 to 3.

After vertical folds, we have a vertical strip from x=1 to 2, y=0 to 3, with three layers stacked: row 1,2,3 all at same x, but different y.

In the vertical fold, we stacked the columns, so for each y, we have three layers of columns.

But the y-coordinates are still from 0 to 3, so the paper is 1 unit wide, 3 units high.

Now, to reduce height, we fold the rows.

For example, fold along y=1. y=1 is between row 1 and 2.

Fold row 1 (y=0 to 1) over row 2 (y=1 to 2) along y=1.

After folding, row 1 is on top of row 2 at y=1 to 2, and row 3 is still from y=2 to 3.

So now, the paper is from y=1 to 3, with at y=1 to 2, two layers (row 1 and 2), at y=2 to 3, one layer (row 3).

Then fold along y=2. y=2 is between row 2 and 3.

After previous, row 1 and 2 are stacked at y=1 to 2, row 3 at y=2 to 3.

Fold along y=2, fold row 3 up over the stack.

So fold row 3 along y=2 to the left (down, since y increasing up or down? Let's assume y increasing up, so folding up or down.

Set y=0 at bottom, y=3 at top.

So row 1: y=0 to 1 (bottom row), row 2: y=1 to 2 (middle), row 3: y=2 to 3 (top).

After vertical folds: paper from x=1 to 2, y=0 to 3, with three layers for each (x,y) point, but since it's a strip, for each y, there are three columns stacked, but the y-range is still 0 to 3.

Now fold rows: fold along y=1. Folding row 1 (y=0 to 1) over row 2 (y=1 to 2) along y=1. Since row 1 is below, folding it up over row 2.

After folding, the point that was at (x,y) for y<1 is now at (x, y+1) or something? When you fold along y=1, the material below y=1 is folded up, so for example, a point (x,0.5) in row 1 is mapped to (x,1.5) if we fold it over, but since we're folding onto row 2, which is from y=1 to 2, so after folding, row 1 is now on top of row 2, so the y-coordinate for row 1 is now from y=1 to 2, but row 2 is also from y=1 to 2, so they are stacked at y=1 to 2.

Similarly, row 3 is still from y=2 to 3.

So the paper now has: from y=1 to 2, two layers (row 1 and row 2), from y=2 to 3, one layer (row 3), and x from 1 to 2.

Width is 1, height from y=1 to 3, so 2 units high.

Then fold along y=2. y=2 is the boundary.

Fold row 3 (y=2 to 3) down over the stack at y=1 to 2.

Folding along y=2, row 3 is above, so fold it down onto the y=1 to 2 area.

After folding, row 3 is now on top of the stack at y=1 to 2, so at y=1 to 2, we have multiple layers: the existing two (row 1 and 2) plus row 3 on top, so three layers at y=1 to 2.

And the paper is now from y=1 to 2, size 1x1, with three layers of columns and three layers of rows, so 9 layers stacked.

Yes, so after this sequence, we have a small square of size 1x1 at the location, say, (x=1 to 2, y=1 to 2), which is the center small square (2,2) in the grid.

In my coordinate, x from 1 to 2 is column 2, y from 1 to 2 is row 2, so small square (2,2).

And it has 9 layers stacked.

Now, the resulting small square is this 1x1 area with 9 layers.

Then, we cut one of the corners of this small square. Since it's a small square, it has four corners.

Let's say we cut the bottom-left corner of this small square, for example. Since it's at (x=1 to 2, y=1 to 2), bottom-left corner is (1,1), top-right (2,2), etc.

Assume the small square has corners: let's say bottom-left (1,1), bottom-right (2,1), top-left (1,2), top-right (2,2), with y=1 bottom, y=2 top.

Cut one corner, say bottom-left corner (1,1).

We cut off that corner, meaning we remove a small piece at (1,1), which affects all 9 layers that are stacked there.

Since it's a corner cut, it might be a small triangle or something, but for simplicity, we can think that we cut out a small region around (1,1).

But when we cut, we are cutting through all layers at that point.

So, at the folded state, we cut a small piece at the corner, which cuts through all 9 layers at that location.

Then, when we unfold, each of the 9 small squares will have a corner cut off, specifically the corner that was at (1,1) in the folded state.

In the folded state, the point (1,1) corresponds to the bottom-left corner of the small square (2,2), but in the original grid, each small square has its own corners.

When we cut at (1,1) in the folded state, which is the bottom-left of the stack, it means that for each layer, the point that was mapped to (1,1) is cut.

In the original paper, which points are mapped to (1,1) during folding?

In the folding sequence, we need to track the mapping.

Since we cut at the corner of the small square, and in the folded state, all layers are aligned, so cutting the corner affects the corner point of each layer.

When unfolded, for each small square, the corner that was at the folded corner will be cut.

In this case, since we folded to (2,2), and cut its bottom-left corner, which is (1,1) in the local coordinate of the small square.

But for the layers, each layer was originally from different positions.

Let's think about the original positions.

In the vertical folding: first, we folded columns.

We folded column 1 over column 2 along x=1. When we did that, the point from column 1 at, say, (0.5, y) for y in 0 to 3, after folding over to column 2, it is now at (1.5, y), because x=0.5 is mapped to x=1.5 when folded.

Similarly, all points in column 1 are mapped to x=1 to 2, but specifically, the left edge x=0 is mapped to x=1, right edge x=1 is mapped to x=1 (the fold line), but since it's folded over, the coordinate is mirrored or something.

When folding along a line, the points are reflected over that line.

So, folding along x=1: the region x<1 is folded over to x>1.

So for a point in column 1, x from 0 to 1, after folding over x=1 to the right, the new x-coordinate is 2 - x, because reflection over x=1.

For example, point at x=0.5, distance to x=1 is 0.5, so after reflection, it is at x=1 + 0.5 = 1.5.

Point at x=0, reflected to x=2? No.

Reflection over x=1: the line x=1 is the mirror.

So for a point at x_a in [0,1), the reflection is at x_b = 2 - x_a, because (x_a + x_b)/2 = 1, so x_b = 2 - x_a.

Similarly, y unchanged.

So for example, point (0.5, y) mapped to (1.5, y)

Point (0, y) mapped to (2, y)

But x=0 is the edge, after mapping to x=2, which is the boundary.

In the paper, after folding, the material from x=0 to 1 is now on top of x=1 to 2, with x-coordinate from 1 to 2, but the values: the point that was at (x,y) for x<1 is now at (2-x, y) in the new coordinate, but since it's stacked, the physical location is at the same (x,y) for x in [1,2], but multiple points.

Perhaps it's better to think in terms of the physical paper.

After folding column 1 over column 2 along x=1, the paper now has the area from x=1 to 3, but at x=1 to 2, there are two layers: the original column 2 and the folded column 1. At x=2 to 3, only column 3.

Then when we fold column 3 over along x=2.

x=2 is the line between x=2 and x=3? x=2 is a grid line.

Folding along x=2: the region x>2 is folded over to x<2.

So column 3, x from 2 to 3, after folding left, the points are reflected over x=2.

Point at x=2.5, reflected to x=1.5? Reflection over x=2.

Point at x_a for x_a >2, new x = 4 - x_a, because (x_a + x_new)/2 = 2, so x_new = 4 - x_a.

For example, x=2.5, x_new = 4 - 2.5 = 1.5

x=3, x_new = 4-3=1

x=2, stays on the fold line.

After folding column 3 over to the left, it is now on top of the area at x=1 to 2.

At x=1 to 2, we had two layers (column 1 and 2), now we add column 3 on top, so three layers at x=1 to 2.

And the y-range is 0 to 3.

Similarly for rows.

Later for rows, we fold along y=1 and y=2.

In the row fold, we had:

After vertical folds, paper from x=1 to 2, y=0 to 3, with three layers.

Then fold along y=1: fold the bottom part y<1 over to the top.

Region y<1, fold over y=1 to y>1.

Reflection over y=1.

Point at (x,y) for y<1, new y = 2 - y.

Because (y + y_new)/2 =1, y_new = 2 - y.

So for example, point at (x,0.5), mapped to (x,1.5)

After folding, row 1 (y=0 to 1) is now on top of row 2 (y=1 to 2) at y=1 to 2.

Then fold along y=2: fold the top part y>2 over to the bottom.

Region y>2, fold down over y=2.

Reflection over y=2.

Point at (x,y) for y>2, new y = 4 - y.

Then after folding, row 3 is on top of the stack at y=1 to 2.

So at the folded state, we have a small square at x=1 to 2, y=1 to 2, with 9 layers.

Now, each layer corresponds to an original small square.

For a point in the folded state at (1,1), which is the bottom-left corner.

In the folded state, (1,1) is a point on the paper.

Which original points are mapped to (1,1)?

First, in the vertical folds: at the folded state, x=1 to 2, y=0 to 3.

But after row folds, we have only y=1 to 2.

But let's find the mapping to original coordinates.

For a point in the folded small square at (x,y) with x in [1,2], y in [1,2].

This point in folded state corresponds to multiple original points due to layering.

For example, the bottom layer in the stack might be the original center, but let's find which original points are at (x,y) in folded state.

From the folding sequence:

The folded state has x from 1 to 2, y from 1 to 2.

This area has 9 layers:

• The layer that is the original row 2, column 2: it was always at x=1 to 2, y=1 to 2, so for any point (x,y) in [1,2]x[1,2], the original point (x,y) for the square (2,2) is itself, but with creases.

Perhaps for a given (x,y) in folded state, it comes from different original points depending on the layer.

But for the cut, we care about the corner.

At the folded corner (1,1).

In the folded state, point (1,1).

Which original points are at this location when folded.

First, through the vertical folding.

In the vertical folding, the point (1,1) in the folded state after vertical folds.

After vertical folds, before row folds, we had the paper from x=1 to 2, y=0 to 3.

At that stage, for a point (x,v) in [1,2]x[0,3], it could be:

• If it is from column 2, it is at original (x,v)

• If from column 1, it is the reflection of (x',v) where x' in [0,1], so original x = 2 - x, because when folded, x in [1,2] corresponds to original x for column 2: x, or for column 1: 2 - x.

Similarly for column 3: after folded, x in [1,2] corresponds to original x for column 3: 4 - x, because reflection over x=2.

For the vertical part:

After folding:

• For the layer from column 1: a point at (x,v) in the vertical strip comes from original (2-x, v) for x in [1,2], v in [0,3]

• For column 2: from original (x,v)

• For column 3: from original (4-x, v) for x in [1,2], v in [0,3]

For example, at x=1, for column 1 layer: original (2-1, v) = (1,v)

For column 2 layer: original (1,v)

For column 3 layer: original (4-1,v) = (3,v)

Similarly at x=2, column 1: (2-2,v) = (0,v) but x=0 is edge, original (0,v) for column 1.

Column 2: (2,v)

Column 3: (4-2,v) = (2,v)

But (2,v) is the line, shared.

Now, for the row folding.

After vertical folds, we have the paper from x=1 to 2, y=0 to 3, with three layers as above.

Then we fold rows.

First, fold along y=1: fold the bottom part (y<1) to the top.

After folding, the paper is from y=1 to 3, with at y=1 to 2, two layers: the original row 2 and the folded row 1.

At this point, for a point (x,y) in [1,2]x[1,2], before the second row fold.

After first row fold (along y=1), the paper is from y=1 to 3.

For y in [1,2], it has:

• The layer from row 2: original (x,y) for y in [1,2]

• The layer from row 1: which is the reflection of (x, y') for y' in [0,1], so original y = 2 - y, because reflection over y=1.

For a point at (x,h) in the range y=1 to 2 after first row fold, for the row 1 layer, it comes from original (x, 2 - h) because when reflected, y' = 2 - h for h in [1,2].

Similarly, for row 2 layer, it is from (x,h) original.

Also, for the row 3, it is still from (x,h) for h in [2,3], but not in this y-range yet.

For the area y=1 to 2 after first row fold, we have two layers: one from row 1 reflected, and one from row 2.

Specifically, at a point (x,y) with y in [1,2]:

• For the row 2 layer: it is from original (x, y) (since row 2 is at y=1 to 2)

• For the row 1 layer: it is from original (x, 2 - y) because the reflection: when y=1, 2-y=1, y=2, 2-y=0, so from y=0 to 1 mapped to y=1 to 2.

Then, we have row 3 from y=2 to 3, but not in this range.

Then, we fold along y=2: fold the top part (y>2) down over y=1 to 2.

Folding along y=2, the region y>2 is folded down.

After folding, row 3 is on top of the stack at y=1 to 2.

For a point at (x,y) in the final folded state, y in [1,2], it has three layers: from row 1, row 2, and row 3.

For row 3, it is the reflection of (x, y') for y' in [2,3], so original y = 4 - y, because reflection over y=2.

For a point at (x,k) in the final y=1 to 2, for the row 3 layer, it comes from original (x, 4 - k) because when reflected over y=2, y' in [2,3] mapped to y=2 to 1, but since we fold down, it's similar.

Reflection over y=2: point at y' >2, new y = 4 - y' for the folded position.

But in the final stack, at (x,y) for y in [1,2], the row 3 layer is from original (x, 4 - y)

Similarly, for row 2 layer, from original (x,y)

For row 1 layer, from original (x, 2 - y)

And for the columns, we have the vertical layers.

So for a point (x,y) in the final folded state, with x in [1,2], y in [1,2], it corresponds to:

• Column 1 layer: original (2-x, v) where v is the row mapping.

But the row mapping depends on which row layer.

Let's combine.

For a point in folded state at (x,y), it can be from:

• Column 1 and row 1: original (2-x, 2-y)

• Column 1 and row 2: original (2-x, y)

• Column 1 and row 3: original (2-x, 4-y)

• Column 2 and row 1: original (x, 2-y)

• Column 2 and row 2: original (x, y)

• Column 2 and row 3: original (x, 4-y)

• Column 3 and row 1: original (4-x, 2-y)

• Column 3 and row 2: original (4-x, y)

• Column 3 and row 3: original (4-x, 4-y)

Now, the corners of the folded small square: for example, the bottom-left corner (1,1).

At (1,1), we cut off the corner, which means we remove a small area around (1,1), affecting all 9 layers.

For each layer, the point (1,1) in folded state corresponds to a point in the original paper that is cut.

For instance, for col1 row1 layer: (2-1, 2-1) = (1,1)

For col1 row2: (2-1,1) = (1,1) y=1, but row2 is y=1 to 2, so (1,1) is the bottom edge.

Similarly, for col1 row3: (2-1,4-1) = (1,3)

Original row 3 is y=2 to 3, so (1,3) is the top-left corner of row 3, but since y=3, it's the top edge.

Let's list the original points that are mapped to (1,1) in folded state.

For each layer:

1. Col1 row1: original (2-x, 2-y) at (1,1): (2-1,2-1) = (1,1)

2. Col1 row2: original (2-x, y) at (1,1): (2-1,1) = (1,1)

3. Col1 row3: original (2-x, 4-y) at (1,1): (2-1,4-1) = (1,3)

4. Col2 row1: original (x, 2-y) at (1,1): (1,2-1) = (1,1)

5. Col2 row2: original (x,y) at (1,1): (1,1)

6. Col2 row3: original (x,4-y) at (1,1): (1,4-1) = (1,3)

7. Col3 row1: original (4-x, 2-y) at (1,1): (4-1,2-1) = (3,1)

8. Col3 row2: original (4-x, y) at (1,1): (4-1,1) = (3,1)

9. Col3 row3: original (4-x,4-y) at (1,1): (4-1,4-1) = (3,3)

Now, the original points:

• (1,1): this is the bottom-left corner of the big square, and it appears in layers: 1,2,4,5

Specifically: col1 row1: (1,1), col1 row2: (1,1), col2 row1: (1,1), col2 row2: (1,1)

But (1,1) is a single point, but in the original grid, it is the corner point of small square (1,1) and also shared.

But when we cut, we cut a small area, so for the corner, it will be cut.

Similarly, (1,3): from col1 row3 and col2 row3

(1,3) is the top-left corner of the big square.

And (3,1): from col3 row1 and col3 row2

(3,1) is bottom-right corner.

And (3,3): from col3 row3, top-right corner.

When we cut the bottom-left corner of the folded small square at (1,1), we are cutting points around (1,1) in the folded state, which correspond to original points like (1,1), (1,3), (3,1), (3,3), but not all at the same time; for different layers, different points.

For example, for the layer col1 row1, we cut at (1,1) which is original (1,1)

For col1 row2, we cut at (1,1) which is original (1,1) — but (1,1) is the same point, but in different layers, it is the same physical point, but in the paper, the point (1,1) is cut multiple times, but it's one point.

I think I'm confusing the point.

The point (1,1) in the big square is a single point, but in the folding, when we have multiple layers, that point might be present in the stack, but for the mapping, when we cut at the folded corner, for each layer, we are cutting a different original point.

From above, for the 9 layers, at folded (1,1), the original points cut are:

• For layer col1 row1: point (1,1)

• For layer col1 row2: point (1,1) -- same point

• For layer col1 row3: point (1,3)

• For layer col2 row1: point (1,1)

• For layer col2 row2: point (1,1) -- again (1,1)

• For layer col2 row3: point (1,3)

• For layer col3 row1: point (3,1)

• For layer col3 row2: point (3,1)

• For layer col3 row3: point (3,3)

So the points that are cut are: (1,1) multiple times, (1,3) twice, (3,1) twice, and (3,3) once.

But (1,1) is cut for layers 1,2,4,5: that is, for the combinations where it is mapped.

When we cut, we remove a small piece, so for the original paper, the points that are cut are: the bottom-left corner (1,1), the top-left corner (1,3), the bottom-right corner (3,1), and the top-right corner (3,3).

Specifically:

• (1,1) is cut for the layers that have it, but since it's a point, it will be cut.

Similarly for others.

But when we unfold, the cut at (1,1) in folded state affects all points that were mapped to it, so in the original, the points (1,1), (1,3), (3,1), (3,3) are cut, but (1,1) is cut for multiple layers, but it's the same point, so it has one hole, but the cut is small, so it's a cutout at the corner.

Perhaps we should think of the cut as removing a small region.

When we cut off the corner of the small square, we remove a small triangle or square from the corner.

For the folded state, at (1,1), we remove a small area, say a square of size ε, so we cut out a small square region around (1,1).

Then, for each layer, we cut out a small area at the point that was at (1,1) in that layer.

From the mapping, for each layer, the point cut is different.

For example:

• For col1 row1: we cut original (1,1)

• For col1 row2: we cut original (1,1) — same location

• For col1 row3: we cut original (1,3)

• For col2 row1: we cut original (1,1)

• For col2 row2: we cut original (1,1)

• For col2 row3: we cut original (1,3)

• For col3 row1: we cut original (3,1)

• For col3 row2: we cut original (3,1)

• For col3 row3: we cut original (3,3)

So the original points that are cut are: (1,1) and (1,3) and (3,1) and (3,3)

Specifically, (1,1) is cut for four layers: but since it's the same point, when we cut, we cut the same physical location multiple times, but in the paper, it's one point, so it will have a cut.

Similarly, (1,3) is cut for two layers: col1 row3 and col2 row3, so at (1,3), it is cut twice, but again, same point.

(3,1) for two layers, (3,3) for one layer.

But the key is that the points (1,1), (1,3), (3,1), (3,3) are cut, which are the four corners of the big square.

(1,1) is bottom-left, (1,3) top-left, (3,1) bottom-right, (3,3) top-right.

When we cut off the corner of the small square, we are cutting off these four corners of the original big square.

But in the folded state, we cut one small area, but due to folding, it corresponds to cutting multiple points.

Specifically, for the corner cut, we are removing a small piece from each of these four corner points.

When unfolded, the big paper will have small pieces missing at each of the four corners.

So there are four holes, one at each corner.

But the problem asks for "holes on the inner area," and "inner area" might not include the corners.

The corners are the boundaries.

Perhaps "inner area" means the area inside the grid, not the boundary.

But the corners are part of the boundary.

Maybe the holes are at the corners, but let's see.

Another thing: when we cut off the corner, it might not be exactly at the point, but a small region, so it creates a hole or a cut at the corner.

But in terms of the paper, after cutting and unfolding, there are four cut-off corners, so the paper has notches at the corners.

But the problem says "holes," which might imply through cuts, but a corner cut might be a hole if it's removed.

Perhaps it's like punching a small hole at the corner.

But the problem says "cut off," so it removes a small piece.

Then, when unfolded, there are four such cut-off regions at the four corners.

But the question is about "holes on the inner area."

What is the inner area? It might not be the corners.

Perhaps for the big square, the inner area is the center region, not the edges.

But the corners are cut, which are not inner.

In this case, with this folding, we have cuts at the corners.

But let's confirm with the mapping.

At the folded corner (1,1), for each layer, we cut a point, but the points cut are (1,1), (1,3), (3,1), (3,3), which are the four corners.

For (1,1), it is cut for multiple layers, but it's the same point, so the cut is deeper or something, but in terms of the paper, it will have a cut at that corner.

Similarly for others.

So after unfolding, the four corners of the big square have a small piece missing.

Now, the problem is to find holes on the inner area.

The inner area might be the area not on the boundary, so the corners are not inner.

Perhaps there are no holes in the inner area, only at the corners.

But that can't be, because the problem asks for the number, implying there are some.

Perhaps with this folding, when we cut, it also affects other points, but at the corner point, only these are mapped.

For example, if we cut a small area around (1,1) in folded state, say a square from (1,1) to (1+ε,1+ε), then for each layer, we cut a small area around the corresponding original point.

For col1 row1: cut around (1,1)

For col1 row2: cut around (1,1) — same area

For col1 row3: cut around (1,3)

For col2 row1: cut around (1,1)

For col2 row2: cut around (1,1)

For col2 row3: cut around (1,3)

For col3 row1: cut around (3,1)

For col3 row2: cut around (3,1)

For col3 row3: cut around (3,3)

So in the original paper, we have cuts around (1,1), (1,3), (3,1), and (3,3).

At (1,1), it is cut for multiple layers, but since it's the same location, it will have a larger cut or something, but essentially, there is a hole or cut at each corner.

But the inner area, if defined as the interior, might have no holes.

Perhaps "inner area" means the area that was folded, like the center square.

Another idea: in the context, "inner area" might refer to the region that was the center of the grid or the squares that are not on the edge.

But in this case, the cuts are at the corners, which are on the edge.

Perhaps for the cut, when we cut the corner of the small square, and it corresponds to the original corners, but the small square we folded to is the center, so its corner is not the original corner.

In our case, we folded to (2,2), and cut its bottom-left corner at (1,1), which is the bottom-left of the center square, but in the original, (1,1) is the corner of the big square, not the center.

The center of the big square is at (1.5,1.5) or something.

But the point (1,1) in absolute coordinates is the corner.

I think I have a mistake.

In the original big square, coordinates: x from 0 to 3, y from 0 to 3.

Small squares: (1,1): x0-1, y0-1; (1,2): x0-1, y1-2; (1,3): x0-1, y2-3; (2,1): x1-2, y0-1; (2,2): x1-2, y1-2; (2,3): x1-2, y2-3; (3,1): x2-3, y0-1; (3,2): x2-3, y1-2; (3,3): x2-3, y2-3.

When we fold to the center, small square (2,2): x1-2, y1-2.

We cut its bottom-left corner: (1,1) in the big coordinate, which is the point (1,1), which is the corner of square (2,2) and also the corner of the big square? (1,1) is the grid point, corner of (1,1), (1,2), (2,1), (2,2).

But in terms of the cut, when we cut at (1,1) for the folded state, as mapped, we cut original points like (1,1), (1,3), etc.

(1,1) is a corner point.

Perhaps for this folding, the cuts are at the corners, so no holes in the inner area.

But let's think about "inner area." Maybe it means the area that was the folding lines or something.

Another common interpretation: in paper folding, when you fold a map, the "inner area" might refer to the area that is covered or the layers that are not the outer.

But in this case, when unfolded, the cuts are at the corners.

Perhaps the problem means that after cutting and unfolding, the paper has holes, and "inner area" might mean the area inside the original grid, but the corners are part of it.

Or perhaps "inner area" means the region that was not the folded edge or something.

I recall that in some such problems, "inner area" refers to the area that was on the inside of the folds, like when you fold, some parts are bent inward.

But in this case, for the 3x3 grid, when we fold to the center, the outer squares are folded over, so they are on the outside or inside?

In the folding process, when we fold the outer columns inward, they are on top, so they might be considered the outside layers, while the center is always inner.

But when we cut, we cut the corner, which is the same for all.

Perhaps for the cut, since we cut at the corner of the small square, and in the original, for the center square, its bottom-left corner is cut, and for other squares, their corners are cut at different positions.

For example, from the mapping, when we cut at (1,1) in folded state, for the center square (2,2), we cut its bottom-left corner at (1,1).

For square (1,1), in the folding, when it was folded, its point (1,1) is mapped to the folded point, but in the cut, for col1 row1, we cut original (1,1), which is the top-right corner of square (1,1)? Let's see.

For small square (1,1): it is from x0-1, y0-1, so its corners are (0,0), (1,0), (0,1), (1,1).

Similarly, (1,1) is the top-right corner of (1,1) if y increasing up.

Assume y=0 at bottom, so for square (1,1): bottom-left (0,0), bottom-right (1,0), top-left (0,1), top-right (1,1).

Similarly, for square (2,2): from x1-2, y1-2, so bottom-left (1,1), bottom-right (2,1), top-left (1,2), top-right (2,2).

When we cut the folded small square at (1,1), which is the bottom-left of (2,2), so for (2,2), we cut its bottom-left corner.

For other squares, when cut in the folded state, we cut different points.

For example, for square (1,1), in the folded state, when it is layered, its point that is at (1,1) in folded state is its top-right corner (1,1), so we cut the top-right corner of square (1,1).

Similarly, for square (1,3): from mapping, for col1 row3, we cut (1,3), which for square (1,3): x0-1, y2-3, so (1,3) is the top-right corner if y=3 top.

y=0 bottom, y=3 top, so for (1,3): bottom-left (0,2), bottom-right (1,2), top-left (0,3), top-right (1,3).

So (1,3) is top-right corner.

Similarly, for (3,1): bottom-left of (3,1) is (2,0), etc., (3,1) is from x2-3, y0-1, so (3,1) is top-right corner if x from 2 to 3, y from 0 to 1, so (3,1) is top-right corner.

Similarly, (3,3) is top-right corner of (3,3).

So in general, for each small square, when we cut the corner in the folded state, for the center square (2,2), we cut its bottom-left corner.

For the corner squares, we cut their top-right corner or something.

But for the cut, in the unfolded paper, each small square will have one corner cut off, but the corner cut is at different corners for each.

For example:

• Square (1,1): top-right corner cut off

• Square (1,2): from mapping, for col1 row2, we cut (1,1) in original, which for (1,2): x0-1, y1-2, so (1,1) is the bottom-right corner? (1,1) is at x=1, y=1, for (1,2), y from 1 to 2, so (1,1) is the bottom-right corner if x from 0 to 1.

Similarly, for (1,2): bottom-left (0,1), bottom-right (1,1), top-left (0,2), top-right (1,2).

So (1,1) is bottom-right corner.

Similarly, for (2,1): from col2 row1, we cut (1,1), for (2,1): x1-2, y0-1, so (1,1) is top-left corner.

And so on.

But the key point is that for the inner area, if "inner area" means the center square, it has a corner cut off.

But the problem says "holes on the inner area," and "holes" might imply something else.

Perhaps when we cut the corner, and since the paper is folded, the cut might create a hole in the center or something.

Another thought: in this folding, when we cut the corner, for the layers that are from the outer squares, the cut is at their corner, but when unfolded, the cut might not be at the edge for the inner area.

But in this case, for the center square, the cut is at its corner, which is on the edge.

Perhaps for a 3x3 grid, the "inner area" is the center square itself.

But it has a corner cut, not a hole.

The problem says "holes," so it might be a through cut or something.

Perhaps the cut is not at the corner, but we cut off a piece, so it creates a hole if it's a separate piece, but when cut off, it's removed, so it's a missing piece, not a hole in the paper.

I think I need to interpret "holes" as the cut-out regions.

But the problem says "holes on the inner area," so we need to see which cut-out regions are in the inner area.

But in this case, with the corner cut, the cut-out regions are at the corners of the big paper, so not inner.

Perhaps for the folding, it is different.

Maybe the folding is to a different small square.

Or perhaps the "resulting small square" is not the center, but a corner.

Let's try that.

Suppose we fold to the bottom-left small square (1,1).

For example, fold column 2 and 3 over onto column 1, etc.

But it might be similar.

Perhaps the cutting is not at the corner of the small square, but the problem says "one of the corners."

Another idea: when we cut off the corner, and unfold, the cut might create a hole in the paper if the cut is through multiple layers in a way that it disconnects, but a corner cut doesn't create a hole; it just notches the corner.

Unless the cut is such that it removes a piece from the interior, but here it's at the corner.

Perhaps for the 3x3 grid, if we fold it in a certain way, the cut corresponds to a hole in the center.

Let's think about the creases.

The dashed lines are the fold lines.

When we fold, the paper is creased, and cutting might add holes.

But in this case, with the cut, it's at the corner.

Perhaps the "inner area" refers to the area bounded by the fold lines, but in a 3x3 grid, the inner area might be the center region.

But it's a grid of squares, so the inner area might be the central part.

I think I found a different interpretation.

In some contexts, for a grid like this, when folded to a small square, the "inner area" might mean the area that was the common point or something.

Perhaps when we cut the corner, and due to the folding, the cut removes a small area that was mapped to the center.

For example, in the folded state, the point (1,1) is cut, but for the center layer, it is the corner, but for other points, it is different.

From the mapping, at (1,1), we cut (1,1), (1,3), (3,1), (3,3), all of which are corners, not the center.

The center point (1.5,1.5) is not cut.

So no hole there.

Perhaps for the cut, we cut a small area, and when unfolded, for the squares that were folded, the cut is at their corner, but for the inner square, it's at the corner.

I think I need to look for a standard solution or think differently.

Another thought: perhaps the folding is not to the center, but to a corner, and we cut the corner, which might correspond to the center or something.

Let's assume we fold to the bottom-left small square (1,1).

So we want to fold the paper so that it becomes size of one small square at (1,1).

For example, first, fold the right part over to the left.

But let's define the sequence.

Suppose we fold along x=1, but x=1 is between column 1 and 2.

Fold column 2 and 3 over onto column 1.

For example, first, fold column 2 over column 1 along x=1.5? The lines are at x=1 and x=2.

Fold along x=2 first. Fold the right part to the left.

Fold along x=2: fold the region x>2 over x=2 to the left.

So column 3 (x2-3) is folded over to x1-2, so column 3 is mapped to x from 1 to 2, but reflected.

Point at x=2.5 to x=1.5, etc.

Then similarly, fold along x=1, but it's messy.

Perhaps fold vertically to the left.

I think it's symmetric to the previous case.

In previous, we folded to center, cut corner, got corner cuts.

Perhaps for the cut, when we cut the corner, and since the paper is folded, the cut might create a hole if it cuts through the fold lines in a certain way.

But I think I should consider that "cut off the corner" means we remove a small square or triangle from the corner, so when unfolded, there are multiple such cuts at different locations.

But in this case, for the 3x3 grid, with 9 layers, we have 9 cuts, but at only 4 distinct locations, with some locations cut multiple times.

But the cuts are at the four corners of the big square.

So after unfolding, the big paper has the four corners cut off.

Now, the "inner area" might be the area excluding the outer boundary, so the corners are not inner, and there are no holes in the inner area.

But that can't be, so perhaps "inner area" means the area that was the folded stack, like the center.

Or perhaps in the context, "inner area" refers to the region that was the bottom layer or something.

Another idea: when the paper is folded into the small square, the "inner area" might mean the area that is on the inside of the fold, but for a flat fold, it's not clear.

Perhaps for the 3x3 grid, after folding to the center, the center square is the inner area, and when we cut the corner, for the center square, it is cut, so it has a corner missing, but not a hole.

The problem says "holes," so it might be that the cut creates a hole in the paper if it's a through cut, but a corner cut is not a through cut; it just rounds the corner.

Unless the cut is made in a way that it removes a piece from the interior, but the corner is not interior.

I think I have a mistake in the folding sequence.

Perhaps the "resulting small square" is not the size of one small square, but after folding, it is a small square shape, but with multiple layers, and we cut one corner, but when cut, if we cut through all layers at the corner, it might create a hole when unfolded if the cut is not at the edge for all.

But in our case, for some layers, the cut is at the corner, which is on the edge.

For example, for the center layer, the cut is at its corner, which is on the edge of the small square, so when unfolded, it's on the edge of the big paper or on the grid line.

But for other layers, the cut is at their corner, which may be inside.

For example, for square (2,2), the cut is at (1,1), which is its bottom-left corner, on the edge.

For square (1,1), the cut is at (1,1), which is its top-right corner, and (1,1) is on the grid point, so it is on the corner of the big paper.

Similarly for all, the cuts are at the grid points or corners.

So no hole in the interior.

Perhaps for the cut, we are cutting a small area, and for the inner squares, the cut is at their corner, which is not on the big paper edge, but on the grid line.

For example, square (2,2) has its bottom-left corner cut, which is at (1,1), on the grid point.

But it's not a hole; it's a cut at the point.

I think I need to search online or think of a different approach.

Perhaps the " holes" refer to the fact that after cutting and unfolding, the cut-out pieces are holes, but they are at the corners.

But the problem says "on the inner area," so perhaps for the big paper, the inner area is the center, and there are no cut-outs there.

Unless with this cutting, it creates a hole in the center from the way the cuts are made.

For example, if we cut the corner, and the paper is folded, the cut might be replicated in a way that it forms a hole.

But in this case, it's not.

Another idea: perhaps the folding is such that the cut is at the corner, but when unfolded, for the creases, the cut might create a tear or something.

I recall that in some paper folding puzzles, when you fold a grid and cut a shape, then unfold, you get a symmetric pattern with holes.

For example, if you fold a square into a small square and cut a circle, you get a circle with multiple layers, but when cut, it's different.

But here, cutting a corner.

Perhaps for a 3x3 grid, if you fold it to the center and cut a corner, when unfolded, you have the big square with the center corner cut, and also the corners of the outer squares cut, but in this case, it's the same as the big square corners.

I think I should consider that the cut removes a small region, and due to folding, it is removed from multiple locations, but those locations are the corners.

So after unfolding, the paper has four small squares missing at the corners, so there are four holes, but at the corners, not inner.

Perhaps the "inner area" is the area within the grid lines, so the corners are included.

But the problem says "inner area," which might imply the interior.

Perhaps in the context, "inner area" means the area that was the folding lines or the center.

Let's assume that for the big square, the inner area is the 1x1 area at the center, from x1 to2, y1 to2.

In that area, is there a hole?

From the cutting, for the point (1.5,1.5), in the folded state, it is at (1.5,1.5), which is not the corner, so not cut.

The cut is at (1,1), so for the inner area, no cut, so no hole.

But for other points, it might be cut, but at (1,1), it is cut, but (1,1) is on the boundary of the inner area.

The inner area is from x1 to2, y1 to2, so (1,1) is a corner point of this area.

If we cut at (1,1), it removes a small piece from the inner area at that corner.

So the inner area has a small hole at its bottom-left corner.

Similarly, for the other cuts, but the other cuts are at (1,3), (3,1), (3,3), which are outside the inner area.

So only at (1,1) for the inner area, but (1,1) is on the boundary.

If we cut a small region around (1,1), it includes points from the inner area.

For example, the point (1,1) and (1+ε,1), (1,1+ε), etc., so for the inner area, the point (1,1) is cut, and (1+ε,1) for small ε, if 1+ε <2, it is in the inner area, so yes, a small part of the inner area is cut.

Similarly, for the cut at (1,3), it is at the top-left, not in the inner area.

So for the inner area, only the bottom-left corner has a small hole.

But the problem asks for "holes," plural, and "on the inner area," so perhaps there is one hole at the corner of the inner area.

But it's not a hole; it's a cut at the corner.

Perhaps it counts as a hole.

But let's see the number.

Perhaps for the cutting, when we cut off the corner, it creates a hole in the paper if the cut is through, but at the corner, it's not a hole; it's an opening.

I think I should consider the number of cut-out regions.

But in this case, with the folding, we have four cut-out pieces, one at each corner, but each cut-out piece is small, so there are four holes at the corners.

But for the inner area, none.

Unless the inner area has the cut at its corner.

But let's assume that the inner area includes its boundary, so there is a hole at the corner.

But it's one point.

I think I need to look for the answer or a different perspective.

Perhaps the folding is done in a way that the small square is not the center, but the grid is folded so that the cut is in the interior.

For example, if we fold the paper in half vertically and horizontally, then we have a smaller square, but for 3x3, it's not integer.

3x3 is odd, so folding might not be to a single square easily.

Another common way is to fold it like a kite or something, but I think the sequence I used is standard.

Perhaps the "dashed lines" include the diagonals or something, but the problem says "along the dashed lines" and the image is not shown, but from the description, it's a grid, so likely only the lines between squares.

Perhaps for the cut, "cut off the corner" means we cut a small square from the corner, so when folded, we remove a small square from the corner of the small square.

Then when unfolded, that small square is removed from multiple locations.

In our case, at the four corners.

So there are four holes at the corners of the big paper.

Then, for the inner area, if it is the center, no holes.

But the problem might consider the corners as part of the inner area or not.

Perhaps "inner area" means the area that was the folded paper minus the outer layers, but it's ambiguous.

I found a similar problem online or in memory.

Perhaps for a 2x2 grid, it would be different, but here it's 3x3.

Let's assume that the folding is such that the cut corresponds to the center.

For example, if we fold to a corner and cut the opposite corner or something.

Suppose we fold to the bottom-left small square (1,1).

Then the resulting small square is at (1,1) size.

We cut its top-right corner, for example.

In the folded state, the small square is from x0 to1, y0 to1, but after folding, it is at that location.

Then cut its top-right corner at (1,1).

Then, which points are cut.

But in the folding, the layers are from the other squares mapped to this area.

For example, to have the paper at the small square (1,1), with multiple layers.

For instance, column 2 folded over column 1, etc.

But it might be messy.

Perhaps the cut at the corner of the small square corresponds to a point that is the center when unfolded.

For example, in the 3x3 grid, the point (1,1) in the folded state at (1,1) might be mapped to (2,2) or something.

In our first mapping, when folded to center, at (1,1) folded, we cut (1,1), (1,3), (3,1), (3,3), not the center.

If we fold to (1,1), then at (1,1) folded, we cut points that map to (1,1).

For example, if we have all paper folded to (1,1), then at (1,1), we cut all layers, so we cut the point (1,1) multiple times, but also other points that are mapped to it.

For instance, if we fold column 2 over column 1 along x=1, then point from column 2 at (1.5,0.5) is mapped to (0.5,0.5) or something.

I think it's complicated.

Perhaps for this problem, the number is 1 or 4.

But let's guess that there is one hole or something.

Perhaps "holes" refer to the number of cut-out regions, and "inner area" means the area inside the cuts or something.

I think I should box the answer as 4, assuming the corners have holes.

But the problem says "inner area," so perhaps it's 0.

Another thought: in the cutting, when we cut off the corner, and unfold, the cut might create a hole in the paper if the cut is made in a way that it cuts through the fold lines, but in this case, it's at the point.

Perhaps for the 3x3 grid, after folding to center, the cut at the corner affects the center point for some layers, but from mapping, not.

Let's list the original points that are cut for the inner area.

The inner area is from x1 to2, y1 to2.

For a point in this area, is it cut?

For example, take (1.5,1.5): in the folded state, it is at (1.5,1.5), which is not (1,1), so not cut.

Take (1,1): it is cut, as we have.

But (1,1) is on the boundary.

Take (1.1,1.1): in the folded state, for the center layer, it is at (1.1,1.1), not cut.

For other layers, it might be mapped to other points.

For example, for col1 row1, the point (1.1,1.1) in original, where is it in folded state?

From the mapping, for col1 row1, the mapping to folded state is x_f = 2-x, y_f = 2-y, so for (1.1,1.1), x_f = 2-1.1 = 0.9, y_f = 2-1.1 = 0.9, which is not (1,1), so not cut.

Similarly for other points in the inner area, they are not at the folded corner, so not cut.

Only the boundary points like (1,1) are cut, but (1,1) is on the boundary of the inner area.

So if we consider the inner area including boundary, there is a cut at (1,1), but it's a point.

I think the problem might have a different folding or interpretation.

Perhaps the "resulting small square" is the size after folding, and we cut its corner, but when cut, if the cut is through all layers, and the corner is not the edge for all, but in the stack, the corner is the edge for all.

I think I should accept that with this folding, there are holes at the four corners, and for the inner area, there are no holes, so the number is 0.

But that seems odd.

Perhaps "inner area" means the area that was the center of the folded stack or something.

Let's read the problem: "how many holes are on the inner area of the piece of paper?"

And "piece of paper" after unfolding.

Perhaps "inner area" is a misnomer, and it means the area inside the original square.

But still.

Another idea: when we cut off the corner, and unfold, the paper has the corners cut, but the cut might create that the center has a hole if the cuts are connected, but with point cuts, it's not.

I think I found a different approach.

Perhaps for the 3x3 grid, when folded to the center, and we cut one corner, the cut removes a small piece that corresponds to the center point for some layers, but from the mapping, at the folded corner, we cut the corner points, not the center.

For example, the center point (1.5,1.5) in the folded state is not cut.

So no.

Perhaps we cut the corner, but the corner of the small square is not the corner of the big paper in the mapping.

In our case, for the center small square, its bottom-left corner is the same as the big paper's bottom-left of the center region.

I think I have to go with the initial reasoning.

Perhaps the number is 1, for the center.

Let's assume that the cut at the folded corner corresponds to the center point when unfolded for some layers.

From the mapping, at (1,1) folded, we cut (1,1), which is not the center.

The center is (2,2) or (1.5,1.5).

(1.5,1.5) is not cut.

So I think for the inner area, there are no holes.

But let's assume that the inner area is the center square, and it has a corner cut, so it has a notch, but not a hole.

The problem says "holes," so perhaps it's 0.

Perhaps after cutting and unfolding, the paper has holes at the fold lines or something.

I give up.

Perhaps the folding is such that the small square is formed by folding the outer parts in, and the cut is at the corner, which when unfolded, creates holes in the inner region.

Let's search for a solution or think logically.

Another thought: in the cutting, when we cut off the corner of the small square, we are removing a small area that, due to the folding, is from the center of the grid for some layers.

For example, in the folded state, the point (1,1) might be the center point for some layer mapping.

From the mapping, for example, for col2 row2, at (1,1) folded, it is (1,1) original, which is not center.

For other, it is different.

For instance, for col2 row2, at (1,1) folded, it is (1,1) original.

For col1 row1, at (1,1) folded, it is (1,1) original.

Same point.

So no.

Perhaps for a different point.

I think I should conclude that there are 4 holes at the corners, and for the inner area, there are none, so the number is 0.

But since the problem asks for it, probably it's not 0.

Perhaps "inner area" means the area that was covered by the small square after folding, which is the center, and it has one hole from the cut.

But the cut is at its corner, so it has a cut corner, not a hole.

Unless the cut is a hole, but it's not; it's a missing piece.

I think I need to box the answer.

I recall that in some sources, for a 3x3 grid folded to center and corner cut, when unfolded, there is one hole in the center or something.

But from mapping, not.

Perhaps the cut is not at the corner of the small square, but the problem says "one of the corners."

Another idea: "cut off the corner" might mean we cut a small square from the corner, so we remove a 1x1 small square area from the corner of the small square.

But the small square is size 1x1, so cutting a small square from its corner would remove a piece of size εxε, but when folded, it removes from multiple.

But in the unfolded, it removes from the corner regions.

So same as before.

I think I have to accept that with this folding, there are holes at the corners, and for the inner area, there are no holes, so the number is 0.

Perhaps the "inner area" is the area that is not the outer layer when folded, but when unfolded, it's all the same.

I think I'll go with 4 for the corners, but since it asks for inner area, perhaps it's 0.

Let's assume that the inner area is the center square, and it has no hole, so 0.

But let's see the answer.

Perhaps for the cut, when we cut the corner, it creates a hole in the paper because the cut goes through the fold lines.

For example, at the corner, the cut might tear along the fold lines.

But that's not reliable.

I think I should look for the number.

Perhaps in the context, "holes" mean the number of cut-out pieces, but the problem says "holes on the inner area," so the holes that are located in the inner area.

In this case, none.

So probably 0.

But let's assume a different folding.

Suppose the folding is not to the center, but to a different square.

Or perhaps the dashing lines are only some, but the problem says "along the dashed lines in some order and direction," so it can be any.

But for the cut to be on the corner, it might be similar.

Perhaps we fold the paper so that the corner cut corresponds to the center.

For example, if we fold the grid in a way that the point (2,2) is mapped to the corner of the small square.

But in the small square, the corner is (1,1) or (2,2) in absolute, but in the folded state, the small square has its own corners.

I think it's impossible to have the center mapped to the corner.

For example, in the folding to center, the center point is at the center of the small square, not at corner.

In folding to a corner, the center point is mapped to somewhere else.

So the center point is never at the corner of the small square.

So when we cut the corner, the center point is not cut.

Therefore, for the inner area, no hole.

So the number is 0.

I think that's the answer.

But let's confirm with a smaller grid.

Suppose a 2x2 grid.

Big square divided to 4 small squares.

Fold along the lines to a small square.

For example, fold to the bottom-left.

Fold vertically along x=1, fold right column over to left, so column 2 is on top of column 1.

Then fold horizontally along y=1, fold top row over to bottom, but after vertical fold, we have from x=0 to1, y=0 to2, with two layers at x=0 to1, y=0 to2 for the row.

After vertical fold: paper from x=0 to1, y=0 to2, with at x=0 to1, y=0 to2, two layers: column 1 and column 2.

Then fold along y=1: fold the top part y>1 over to the bottom.

Fold row 2 over row 1 along y=1.

After folding, row 2 is on top of row 1 at y=0 to1, so the small square is at x=0 to1, y=0 to1, with 4 layers.

Cut bottom-left corner at (0,0).

Then for each layer, at (0,0) folded, the original points:

For col1 row1: (0,0)

For col1 row2: (0,0) for the point, but when folded, for row2 layer, at (0,0) folded, it is from original (0,2) or something.

Mapping:

After vertical fold: for a point (x,y) in [0,1]x[0,2], for column 1 layer: original (x,y)

For column 2 layer: original (2-x,y) if x from 0 to1, but for column 2, x from 1 to2, so at (x,y) for x in [0,1], for column 2 layer, it is from (2-x,y)

Then for row fold: fold along y=1, fold y>1 down.

For a point at (x,y) in [0,1]x[0,1] after both folds.

For the layer from row 1: it is from original (x,y) for y in [0,1]

For row 2 layer: from original (x,2-y) for y in [0,1]

And for columns:

For col1: at (x,y), for row1: (x,y), for row2: (x,2-y)

For col2: at (x,y), for row1: (2-x,y), for row2: (2-x,2-y)

At folded (0,0):

For col1 row1: (0,0)

For col1 row2: (0,2-0) = (0,2)

For col2 row1: (2-0,0) = (2,0)

For col2 row2: (2-0,2-0) = (2,2)

So cut points: (0,0), (0,2), (2,0), (2,2), the four corners of the 2x2 grid.

After unfolding, corners cut off.

For the inner area, if it is the center, no hole.

So for 2x2, also corners.

So probably for any grid, cutting the corner when folded to a small square cuts the corners of the big paper.

So for the inner area, no holes.

Therefore, for this problem, on the inner area, there are 0 holes.

So the answer should be 0.

I think that's it.