I have this problem about a square garden divided into four parts: a pool (P), a flowerbed (F), a lawn (L), and a sandpit (S). The lawn and the flowerbed are both square-shaped. The perimeter of the lawn is 20 meters, and the perimeter of the flowerbed is 12 meters. I need to find the perimeter of the pool.

First, I should probably sketch this out based on the description. The picture shows a big square divided into four smaller regions. From the text, it's arranged with P and F on the top row, and L and S on the bottom row. P is on the top-left, F on the top-right, L on the bottom-left, and S on the bottom-right.

Since the garden is square, let's denote the side length of the big square as $s$ meters.

Now, the lawn (L) and the flowerbed (F) are both squares. Their perimeters are given: lawn has perimeter 20 m, so each side of the lawn is $\frac{20}{4}=5$ m. Similarly, flowerbed has perimeter 12 m, so each side is $\frac{12}{4}=3$ m.

The lawn is in the bottom-left, and it's a square, so its side is 5 m. Similarly, the flowerbed is top-right, side 3 m.

Now, the pool is top-left, and sandpit is bottom-right. But I don't know their shapes yet. The problem doesn't say that P and S are squares, only L and F are.

The garden is divided into four parts, so the divisions might not be equal; there are lines separating them.

Let me think about the dimensions.

Assume the big square has side s.

Let the horizontal division line be at height h from the bottom, so the bottom row (L and S) has height h, and the top row (P and F) has height s - h.

Similarly, for the vertical division, let the left division be at width a from the left, so the left column (P and L) has width a, and the right column (F and S) has width s - a.

Now, the lawn L is in the bottom-left, so it has width a and height h, and since it's a square, we must have a = h.

Similarly, the flowerbed F is in the top-right, so it has width s - a and height s - h, and since it's a square, s - a = s - h.

F is top-right, so its width is from the right edge, so width (s - a), and height from the top, so height (s - h). And since it's square, (s - a) = (s - h).

Similarly, for L, bottom-left, width a, height h, and it's square, so a = h.

From L: a = h

From F: s - a = s - h, which implies a = h, same thing.

s - a = s - h implies -a = -h, so a = h.

Yes, so a = h.

Let me set a = h = x, for example.

So, the width of the left part is x, height of bottom part is x.

Now, since L is square with side x, and perimeter 20, so 4x = 20, thus x = 5.

Perimeter of L is 20, so 4 * side = 20, side = 5, so x = 5.

Similarly, for F, it is square with side, let's say y, perimeter 12, so 4y = 12, y = 3.

Now, F is top-right, so its width is s - a, height is s - h.

Since a = h = x = 5, so width of F is s - 5, height of F is s - 5? Height s - h = s - 5.

And since F is square, s - 5 must equal the side, which is 3? No.

F has side y = 3.

So, s - a = s - 5 = width of F

s - h = s - 5 = height of F

And since F is square, width = height, so s - 5 = y = 3?

That can't be, because if s - 5 = 3, then s = 8.

But let's see.

s - 5 must equal the side of F, which is 3.

So s - 5 = 3, thus s = 8.

Then the big square has side 8 m.

Now, what about the pool and sandpit?

Pool P is top-left, so it has width a = 5, height h_top? Height from top, but the top row has height s - h = 8 - 5 = 3.

Since h = 5, s - h = 3.

P is top-left, so width a = 5, height s - h = 3.

But is P a rectangle? 5 by 3.

Similarly, sandpit S is bottom-right, width s - a = 8 - 5 = 3, height h = 5.

So S is 3 by 5 rectangle.

The question is to find the perimeter of the pool.

Pool P is top-left, width 5, height 3, so it's a rectangle, not square.

Perimeter of P: since it's a rectangle with sides 5 and 3, perimeter is 2*(5 + 3) = 2*8 = 16 m.

But is that the answer? Let me see if this makes sense.

We also have the sandpit S, but we don't care about it.

The garden is square, side 8.

Lawn L: bottom-left, 5x5, area 25, perimeter 20, good.

Flowerbed F: top-right, 3x3, area 9, perimeter 12, good.

Pool P: top-left, 5 wide, 3 high, so 5x3 rectangle.

Sandpit S: bottom-right, 3 wide, 5 high, 3x5 rectangle.

Now, the divisions: vertical line at x=5 from left, horizontal line at y=5 from bottom? Let's set coordinates.

Assume big square from (0,0) to (8,8).

Then vertical divider at x=5, horizontal divider at y=5? h is height of bottom, so from bottom to horizontal divider is h=5, so divider at y=5? If bottom is y=0, top y=8.

Set y=0 at bottom, y=8 at top.

Horizontal divider at height h from bottom, so at y = h = 5.

Similarly, vertical divider at x = a = 5 from left.

So left column: x from 0 to 5, y from 0 to 8? No.

The horizontal divider splits into top and bottom.

Bottom part: y from 0 to h = 5, x from 0 to s=8.

But within that, L is from x=0 to a=5, y=0 to h=5, so [0,5]x[0,5] for L.

Then S is bottom-right: x from a=5 to s=8, y from 0 to h=5, so [5,8]x[0,5].

Top part: y from h=5 to s=8, x from 0 to s=8.

P is top-left: x from 0 to 5, y from 5 to 8, so [0,5]x[5,8]

F is top-right: x from 5 to 8, y from 5 to 8, so [5,8]x[5,8]

F is [5,8]x[5,8], which is 3x3, good.

P is [0,5]x[5,8], which is width 5, height 3, since y from 5 to 8 is 3 units.

Perimeter of P: it has two sides of length 5 (left and right? No, the sides.

As a rectangle, it has two horizontal sides of length 5 each, and two vertical sides of length 3 each.

So perimeter: 25 + 23 = 10 + 6 = 16.

Since it's a rectangle, perimeter 2*(length + width) = 2*(5 + 3) = 16.

Now, the problem is to find perimeter of pool, which is 16.

But is this correct? The garden is divided, and P is not necessarily a square, which it isn't.

But the question is just to find its perimeter.

We have all we need.

But let me see the sandpit S: [5,8]x[0,5], width 3, height 5, perimeter 2*(3+5)=16 similarly.

But not needed.

Is there any issue?

The problem says "the lawn and the flowerbed are of a square shape", which we used, and perimeters given.

In this case, with s=8, it all works.

But is the pool a square? No, but it doesn't have to be.

Perhaps I should consider if the divisions are such that P or S could be squares, but in this case, not.

Another thing: in the arrangement, P and F are on top, but P has width 5, height 3, while F has width 3, height 3, so they are different.

But that should be fine.

Perhaps the vertical and horizontal divisions are not aligned, but in the picture, it seems like a grid, so I think it's fine.

Maybe I need to consider the perimeter including the dividers or something, but no, the perimeter of the pool should be its own boundary.

The pool P shares boundaries with other parts, but when we say perimeter of the pool, it should be the outer perimeter, I think, meaning the boundary that separates it from the surrounding, not including the internal dividers.

Let me read the question.

"the perimeter of the lawn is 20 m", similarly for others.

For the lawn, it is a separate region, so its perimeter should be the length of its boundary, which for a square, is just the four sides, but since it's adjacent to other regions, the internal sides are shared, but when we measure perimeter of the lawn, it should include all its sides, even if shared, because perimeter is the total length around it.

In terms of the region itself, the perimeter is the length of its boundary, regardless of whether it's internal or external.

For example, if you have a square tile, its perimeter is 4 times side, even if you place it next to another.

Similarly here, for the lawn, since it's a square of 5x5, its perimeter is 20, which includes the sides that are shared with pool and sandpit.

Similarly for flowerbed.

For the pool, we need to find its perimeter, which would be the length around it.

In our case, pool is a rectangle 5 by 3, so it has four sides: two of length 5 (top and bottom? Let's see.

Pool P: from x=0 to 5, y=5 to 8.

So, left side: x=0, y from 5 to 8, length 3

Bottom side: y=5, x from 0 to 5, but y=5 is the internal horizontal line, so this side is shared with the lawn below.

Similarly, right side: x=5, y from 5 to 8, shared with flowerbed.

Top side: y=8, x from 0 to 5, which is the top edge of the garden.

Similarly, for the perimeter, we include all four sides.

So sides:

• Left: vertical, length 3, from (0,5) to (0,8)

• Bottom: horizontal, from (0,5) to (5,5), length 5, shared with lawn

• Right: vertical, from (5,8) to (5,5), length 3, shared with flowerbed? From (5,5) to (5,8), that's upward, but y from 5 to 8, so from (5,5) to (5,8), length 3.

• Top: horizontal, from (0,8) to (5,8), length 5, part of the garden boundary.

So perimeter: left 3 + bottom 5 + right 3 + top 5 = 3+5+3+5=16.

As a rectangle, 2*(width + height) = 2*(5+3)=16.

Similarly, for the lawn: it has bottom and left as garden boundary, top and right shared.

But perimeter is still 4*5=20, including shared sides.

So for pool, it should be fine.

But in this case, we have perimeter 16.

But is this correct? I think I might have a mistake.

The flowerbed F is 3x3, but in the top-right, with s=8, a=5, so s-a=3, s-h=3, so F is 3x3.

But what about the pool? It has width 5, height 3.

But the problem is to find perimeter of pool, which is 16.

But let me see if there is another possibility.

Perhaps the divisions are not with vertical and horizontal lines, but the picture shows it as a grid, I think.

The picture is given, but since I don't have it, I assume it's divided by straight lines horizontally and vertically.

Perhaps P and S are not rectangles, but in this case, they are.

Another thing: the sandpit S is 3x5, which is not square, but that's fine.

But let me confirm the perimeters.

Lawn L: 5x5 square, perimeter 20, good.

Flowerbed F: 3x3, perimeter 12, good.

Pool P: 5x3 rectangle, perimeter 2(5+3)=16.

But the question is perimeter of pool, so 16 meters.

But I feel like maybe I missed something because the pool is not a square, and the perimeter is not given directly.

Perhaps I need to consider that the garden is square, and find the side.

Another thought: in my calculation, I have s=8, but let me denote variables.

Set side of lawn as L_s = 5

Side of flowerbed as F_s = 3

Now, from the division, since L is bottom-left square, it occupies a square of 5x5 in the corner.

Similarly, F is top-right square, 3x3.

Now, for F to be in the top-right, the distance from right must be F_s, and from top must be F_s.

Similarly for L, from left and bottom.

Now, let the big square side be S.

Then, the left edge to the vertical divider is the width of L, which is L_s = 5.

Similarly, the bottom to horizontal divider is height of L, which is also L_s = 5, since it's square.

Similarly, for F, the distance from right edge to vertical divider should be F_s = 3? No.

Let the vertical divider be at position A from left, so L has width A, and since square, height also A for the bottom part? Not necessarily.

Earlier I had the horizontal divider at height H from bottom.

Then L has width A and height H, and since square, A = H.

Similarly, F has width S - A (from divider to right) and height S - H (from divider to top), and since square, S - A = S - H, so A = H.

Thus A = H = let's say X.

Then L has side X, so X = L_s = 5.

Similarly, F has side S - X, so S - X = F_s = 3.

Thus S - 5 = 3, S = 8.

Then P has width A = 5, height S - H = S - X = 3, so 5 by 3.

Perimeter 2(5+3)=16.

If I think in terms of the pool.

The pool is adjacent to L on bottom, but in terms of size.

Perhaps the pool shares the left side with garden, but not necessarily.

I think it's consistent.

But let me see the answer.

The problem is from a source that might expect a boxed answer.

But 16 seems straightforward, but I recall that in the beginning, I thought it might be tricky.

Another thing: the sandpit S is in the bottom-right, and it has width S - A = 3, height H = 5, so 3x5, which is not square, but that's fine.

Similarly, pool is not square.

But the perimeter is asked for pool.

Perhaps I misread the positions.

The picture shows P, F on top, L, S on bottom.

P top-left, F top-right, L bottom-left, S bottom-right.

In my setup, with S=8, P is 5x3, which is wider than tall, while L is 5x5, F is 3x3, S is 3x5.

Now, the pool perimeter is 16.

But let me calculate the actual perimeter length.

As I listed: left side: 3m, bottom side: 5m (shared with L), right side: 3m (shared with F), top side: 5m (garden edge).

Total: 3+5+3+5=16m.

If I consider the shared sides, but for the region, the perimeter includes all, so it should be 16.

Perhaps for the pool, the bottom and right are internal, but we still include them in the perimeter measurement.

I think it's correct.

Maybe the problem means the outer perimeter when it's part of the garden, but no, the perimeter of the pool is for the pool area itself.

I think I should proceed.

But let me see if there is another interpretation.

Suppose that the vertical and horizontal divisions are not at the same point, but in the picture, it should be a grid.

Perhaps P and S are not rectangles, but in this division, they are.

Another idea: perhaps the lawn and flowerbed are squares, but not necessarily aligned with the corners in that way, but the picture shows they are in corners.

L is bottom-left, F top-right, so corners.

So I think it's fine.

Perhaps the pool or sandpit could be squares, but in this case, with given perimeters, it doesn't work.

For example, if I assume that the vertical divider is not at the side of L.

Suppose that L is square, but the vertical division is not at its right edge.

But in the standard grid division, it should be.

I think I'm overcomplicating.

Perhaps from the perimeters, we can set up equations.

Let the side of big square be S.

Let the width of the left column be A, height of bottom row be B.

Then L has width A, height B, and since square, A = B.

Perimeter of L: 4A = 20, so A = 5.

Similarly, F is in top-right, so it has width S - A, height S - B.

Since A = B, let A = B = 5.

Then F has width S - 5, height S - 5.

And F is square, so S - 5 must equal the side, but perimeter of F is 12, so 4*(S - 5) = 12? No.

The perimeter of F is the length around it.

F is a square of side (S - 5), so perimeter 4*(S - 5) = 12.

Thus 4(S - 5) = 12

S - 5 = 12/4 = 3

S - 5 = 3

S = 8

Same as before.

Then P has width A = 5, height S - B = S - 5 = 3, so rectangle 5x3, perimeter 2(5+3)=16.

If I don't assume A=B, but from L square, A must equal B, because it's a square region.

Similarly for F.

So I think it's consistent.

Therefore, the perimeter of the pool is 16 meters.

But the problem is to box the answer.

So, I think it's 16.

But let me see the date: 2025, but that shouldn't matter.

Perhaps units, but in meters.

So, perimeter is 16.

But I recall that in some similar problems, sometimes the internal perimeters are considered differently, but I don't think so.

Another thought: when we say "perimeter of the lawn", it might be misinterpreted, but I think it's clear.

Perhaps for the pool, since it has two internal sides, but the perimeter length is still the same.

I think I should go with 16.

So, the perimeter of the pool is 16 meters.