Okay, so I need to find the shortest distance that can be traveled starting from any of the points A, B, C, or D, and visiting the other three points once each. Hmm, this sounds like a Traveling Salesman Problem (TSP) where I have to visit all four points with the shortest possible route. But since there are only four points, maybe I can list all possible permutations and calculate the distances to find the minimum. Let me think through this step by step.

First, let me note down the coordinates of each point as given in the Asymptote code:

- A is at (-3, 3)
- B is at (3, 0)
- C is at (0, -4)
- D is at (0, 0)

The Asymptote code also draws lines between these points with the following distances labeled:

- D to B: 3 units
- A to B: 6 units
- A to C: 6 units
- D to C: 4 units
- A to D: 5 units
- C to B: 5 units

Wait, but maybe those labels are just for the diagram, and the actual coordinates might not exactly match those distances. Let me verify the distances between each pair using coordinates.

Calculating the distance between two points (x1, y1) and (x2, y2) is done using the formula sqrt[(x2 - x1)^2 + (y2 - y1)^2].

Let me compute all pairwise distances:

1. A to B: sqrt[(3 - (-3))^2 + (0 - 3)^2] = sqrt[(6)^2 + (-3)^2] = sqrt[36 + 9] = sqrt[45] = 3*sqrt(5) ≈ 6.708. But the diagram says 6 units. Hmm, maybe the diagram is approximate, but the coordinates are exact. So perhaps the distances are as per the coordinates. Therefore, the labels in the diagram might not be accurate. Alternatively, maybe the Asymptote code is using a different scale. Wait, the Asymptote code draws D--B with label "3", but according to coordinates, the distance is sqrt[(3)^2 + (0 - 0)^2] since D is at (0,0) and B is at (3,0). So distance from D to B is 3 units, which matches the label. Similarly, D to C is sqrt[(0 - 0)^2 + (-4 - 0)^2] = 4 units, which matches the label. A to D: sqrt[(-3 - 0)^2 + (3 - 0)^2] = sqrt[9 + 9] = sqrt[18] ≈ 4.242, but the label says 5. Hmm, that's a discrepancy. Similarly, A to B: sqrt[(3 - (-3))^2 + (0 - 3)^2] = sqrt[36 + 9] = sqrt[45] ≈ 6.708, but label says 6. A to C: sqrt[(0 - (-3))^2 + (-4 - 3)^2] = sqrt[9 + 49] = sqrt[58] ≈ 7.616, but label says 6. C to B: sqrt[(3 - 0)^2 + (0 - (-4))^2] = sqrt[9 + 16] = 5 units, which matches the label. So the labels in the diagram are approximate; the actual distances based on coordinates are different. Therefore, the distances we should use are those calculated from coordinates. So the diagram is just a rough sketch, but the coordinates are precise. Therefore, I should use the actual coordinates to compute the distances.

Therefore, the distances between the points are:

- A to B: sqrt[45] ≈ 6.708
- A to C: sqrt[58] ≈ 7.616
- A to D: sqrt[18] ≈ 4.242
- B to C: 5
- B to D: 3
- C to D: 4

Wait, but in the diagram, the distances between A-B, A-C, A-D, B-C, B-D, C-D are labeled as 6, 6, 5, 3, 4, 5 respectively. However, according to coordinates, these distances are different. So perhaps the question is using those labeled distances from the diagram, not the coordinates. Hmm, this is confusing. Let me check again.

The problem says: "What is the shortest distance that can be travelled if you start at any point, A, B, C, or D, and visit the other three points once?"

The Asymptote code draws the figure with specific coordinates and labels the edges with certain lengths. So perhaps the distances are as per the diagram, not the coordinates. Therefore, even though the coordinates might not match the distances, the actual problem uses the labeled distances in the diagram. So maybe the coordinates are just for drawing, but the distances are the ones labeled. Let me check the coordinates again.

Wait, in the Asymptote code, it labels "3" for D-B, "6" for A-B, "6" for A-C, "4" for D-C, "5" for A-D, and "5" for C-B. So the diagram has edges with those lengths. Therefore, even though the coordinates are given, the distances between points are the ones labeled. Therefore, the correct distances to use are:

- A to B: 6
- A to C: 6
- A to D: 5
- B to C: 5
- B to D: 3
- C to D: 4

Therefore, the coordinates are probably just for drawing the figure, but the actual distances to use are the ones labeled. Therefore, regardless of the coordinates, the distances between the points are as per the labels. So the distances are:

AB = 6, AC = 6, AD = 5, BC = 5, BD = 3, CD = 4.

Wait, but in the Asymptote code, the coordinates are:

A = (-3,3); B = (3,0); C = (0,-4); D = (0,0).

So the distance from A to B is sqrt[(3 - (-3))² + (0 - 3)²] = sqrt[6² + (-3)²] = sqrt[36 + 9] = sqrt[45] ≈ 6.708, but the label says 6. Similarly, A to C is sqrt[3² + (-7)²] = sqrt[9 + 49] = sqrt[58] ≈ 7.616, but the label says 6. So there's a conflict here. Therefore, the problem is likely using the distances labeled in the diagram, not the actual coordinates. Therefore, the distances between the points are as per the diagram: AB=6, AC=6, AD=5, BC=5, BD=3, CD=4.

Therefore, the graph has these edge weights:

Between A and B: 6

Between A and C: 6

Between A and D: 5

Between B and C: 5

Between B and D: 3

Between C and D: 4

Therefore, the problem reduces to finding the shortest possible route that starts at any of the four points, visits the other three exactly once, and returns to the starting point? Wait, no. Wait, the question says "visit the other three points once." So it's a Hamiltonian path, not a cycle. So starting at a point, visiting the other three once each, and the total distance is the sum of the edges taken. So it's a path, not a cycle. Therefore, the problem is to find the shortest possible path that starts at any of the four points, visits the other three exactly once, and ends at any of the four points (but since it's a path, not a cycle, it doesn't have to return to the start). Wait, but the problem says "start at any point, A, B, C, or D, and visit the other three points once." So it's a path of four points, starting at one, visiting the other three once each, so total of three edges. Therefore, the total distance is the sum of three edges. Therefore, the problem is to find the shortest possible path (sum of three edges) that visits all four points, starting at any point.

Wait, but if you start at a point, you need to visit the other three, so the path has four points in total (including the start), but the number of edges is three. Therefore, the total distance is the sum of three edges. So we need to find the shortest possible sum of three edges that connect four points, starting at any of them. However, since the path is a sequence of edges, the order matters. So we need to find the shortest path that starts at any point, goes through the other three exactly once, and the total distance is the sum of the three edges.

Therefore, the problem is equivalent to finding the shortest Hamiltonian path in the graph, starting at any vertex. The graph has four vertices with the given edge weights.

Since there are only four points, we can list all possible Hamiltonian paths (starting at each point, going through the others in some order) and calculate their total distances, then choose the minimum.

There are 4 starting points, and for each starting point, 3! = 6 possible orders to visit the other three points. However, since it's a path, some orders may result in the same total distance. Alternatively, since the graph is undirected, the reverse paths will have the same distance.

But given that the graph is undirected, but the distances are as per the diagram, we can model it as a complete graph with four nodes and the given edge weights.

Therefore, the strategy is:

1. For each starting point (A, B, C, D), generate all possible permutations of the other three points. Each permutation represents a possible path starting at the starting point and visiting the other three in that order.

2. For each permutation, calculate the sum of the distances between consecutive points.

3. Find the permutation with the minimum total distance for each starting point.

4. Among all starting points and their minimum total distances, choose the smallest one as the answer.

Alternatively, since the problem allows starting at any of the four points, we can consider all possible starting points and all possible paths, compute their total distances, and pick the overall minimum.

But since there are 4 starting points, each with 6 permutations, that's 24 possible paths. However, since the graph is undirected, some paths are repeats in reverse. For example, starting at A, going B-C-D is the same as starting at D, going C-B-A, but the total distance will be the same. Therefore, we can reduce the number of computations by considering each unique path once.

But since we have only 24 paths, it's feasible to compute them all. Let me list all possible paths and their total distances.

But maybe there is a smarter way. Let's see.

First, note that the edge weights are:

AB=6, AC=6, AD=5, BC=5, BD=3, CD=4.

So, the edges from each node:

From A: AB=6, AC=6, AD=5

From B: BA=6, BC=5, BD=3

From C: CA=6, CB=5, CD=4

From D: DA=5, DB=3, DC=4

Now, let's list all possible paths starting from each node.

Starting from A:

1. A -> B -> C -> D

Distance: AB + BC + CD = 6 + 5 + 4 = 15

2. A -> B -> D -> C

Distance: AB + BD + DC = 6 + 3 + 4 = 13

3. A -> C -> B -> D

Distance: AC + CB + BD = 6 + 5 + 3 = 14

4. A -> C -> D -> B

Distance: AC + CD + DB = 6 + 4 + 3 = 13

5. A -> D -> B -> C

Distance: AD + DB + BC = 5 + 3 + 5 = 13

6. A -> D -> C -> B

Distance: AD + DC + CB = 5 + 4 + 5 = 14

So, the minimum distance starting from A is 13 (paths 2, 4, 5).

Starting from B:

1. B -> A -> C -> D

Distance: BA + AC + CD = 6 + 6 + 4 = 16

2. B -> A -> D -> C

Distance: BA + AD + DC = 6 + 5 + 4 = 15

3. B -> C -> A -> D

Distance: BC + CA + AD = 5 + 6 + 5 = 16

4. B -> C -> D -> A

Distance: BC + CD + DA = 5 + 4 + 5 = 14

5. B -> D -> A -> C

Distance: BD + DA + AC = 3 + 5 + 6 = 14

6. B -> D -> C -> A

Distance: BD + DC + CA = 3 + 4 + 6 = 13

So, the minimum distance starting from B is 13 (paths 6).

Starting from C:

1. C -> A -> B -> D

Distance: CA + AB + BD = 6 + 6 + 3 = 15

2. C -> A -> D -> B

Distance: CA + AD + DB = 6 + 5 + 3 = 14

3. C -> B -> A -> D

Distance: CB + BA + AD = 5 + 6 + 5 = 16

4. C -> B -> D -> A

Distance: CB + BD + DA = 5 + 3 + 5 = 13

5. C -> D -> A -> B

Distance: CD + DA + AB = 4 + 5 + 6 = 15

6. C -> D -> B -> A

Distance: CD + DB + BA = 4 + 3 + 6 = 13

So, the minimum distance starting from C is 13 (paths 4, 6).

Starting from D:

1. D -> A -> B -> C

Distance: DA + AB + BC = 5 + 6 + 5 = 16

2. D -> A -> C -> B

Distance: DA + AC + CB = 5 + 6 + 5 = 16

3. D -> B -> A -> C

Distance: DB + BA + AC = 3 + 6 + 6 = 15

4. D -> B -> C -> A

Distance: DB + BC + CA = 3 + 5 + 6 = 14

5. D -> C -> A -> B

Distance: DC + CA + AB = 4 + 6 + 6 = 16

6. D -> C -> B -> A

Distance: DC + CB + BA = 4 + 5 + 6 = 15

So, the minimum distance starting from D is 13 (paths 3, 4).

Therefore, the minimum distance from each starting point is 13. However, let's verify if these paths are indeed valid and if there are no shorter paths.

Wait, but let's check if there are any other paths that might have a shorter total distance. For example, maybe a different order of visiting the points could result in a shorter total distance. But from the above computations, all possible permutations have been considered for each starting point, so the minimum for each starting point is 13. Therefore, the overall minimum is 13.

But wait, let me confirm the paths again.

Starting from A:

Path 2: A->B->D->C: AB=6, BD=3, DC=4. Total: 6+3+4=13

Path 4: A->C->D->B: AC=6, CD=4, DB=3. Total: 6+4+3=13

Path 5: A->D->B->C: AD=5, DB=3, BC=5. Total:5+3+5=13

All three paths sum to 13.

Starting from B:

Path 6: B->D->C->A: BD=3, DC=4, CA=6. Total:3+4+6=13

Starting from C:

Path 4: C->B->D->A: CB=5, BD=3, DA=5. Total:5+3+5=13

Path 6: C->D->B->A: CD=4, DB=3, BA=6. Total:4+3+6=13

Starting from D:

Path 3: D->B->C->A: DB=3, BC=5, CA=6. Total:3+5+6=14

Wait, but earlier I thought the minimum was 13 for D, but in path 3, it's 14. Wait, let's check the starting from D paths again.

Starting from D:

1. D->A->B->C: DA=5, AB=6, BC=5: 5+6+5=16

2. D->A->C->B: DA=5, AC=6, CB=5: 5+6+5=16

3. D->B->A->C: DB=3, BA=6, AC=6: 3+6+6=15

4. D->B->C->A: DB=3, BC=5, CA=6: 3+5+6=14

5. D->C->A->B: DC=4, CA=6, AB=6: 4+6+6=16

6. D->C->B->A: DC=4, CB=5, BA=6: 4+5+6=15

So, the minimum for D is 14, achieved by path 4: D->B->C->A: 3+5+6=14. Wait, but earlier I thought the minimum was 13. Wait, that contradicts. So, perhaps I made a mistake in the earlier analysis.

Wait, if starting from D, the minimum path is 14, but starting from other points, the minimum is 13. Therefore, the overall minimum is 13. Therefore, the answer is 13.

But let me confirm again.

Starting from D, the path D->B->C->A: DB=3, BC=5, CA=6. Total:3+5+6=14.

Another path: D->B->A->C: DB=3, BA=6, AC=6. Total:3+6+6=15.

Another path: D->C->B->A: DC=4, CB=5, BA=6. Total:4+5+6=15.

But earlier, starting from A, B, C, the minimum was 13. Therefore, the overall minimum is 13.

But wait, let me check the starting from D again. Maybe there's a different path. Wait, perhaps D->A->C->B: DA=5, AC=6, CB=5. Total:5+6+5=16. Not better.

Alternatively, D->A->B->C: 5+6+5=16. Not better.

Therefore, starting from D, the minimum is 14, which is higher than 13.

Therefore, the overall minimum is 13.

Therefore, the answer is 13.

But let me check once more to ensure I haven't missed any other path.

Starting from A:

A->B->D->C: 6+3+4=13

A->C->D->B:6+4+3=13

A->D->B->C:5+3+5=13

Starting from B:

B->D->C->A:3+4+6=13

Starting from C:

C->B->D->A:5+3+5=13

C->D->B->A:4+3+6=13

Starting from D:

D->B->C->A:3+5+6=14

Therefore, the minimum is indeed 13, achieved by multiple paths.

Therefore, the shortest distance is 13.

But let me verify once again if all steps are correct.

Yes, each path starting from A, B, C has a minimum of 13, while starting from D, the minimum is 14. Therefore, the overall minimum is 13.

Therefore, the answer is 13.

**Final Answer**
The shortest distance that can be travelled is $\boxed{13}$.
Therefore, the shortest path that starts at any point $A$, $B$, $C$, or $D$, and visits the other three points once is $\boxed{13}$.
</think>