Equal flow distribution to 3x3 grid

[syphex]

This may be more of a math problem; it arose out of my curiosity of manipulating LED matrices and reminds me of a traffic flow problem, but I am sure it can model a variety of applications. Its not a homework problem but would probably make a good one!

You have a 3x3 grid of points, each point must be provided with an equal amount of flow from one of the outer points. I thought you could "split" the lines from the outer point for 2, 4, 8 (flow in = flow out and the flow is being halved each time), but this only seems to work for a 2x4 configuration, since otherwise the paths to the source would be different and hence flow rate would be effected.

Is this possible? Any advice?

Thanks in advance any help is greatly appreciated.

 

[mfb]

"Flow" of what? What determines flow rate? How do you define "flow rate of a point"?

you could "split" the lines from the outer point for 2, 4, 8

What does that mean?

 

[syphex]

The flow from any arbitary point except for the middle must be distributed equally to the remaining 8 points. Water, electricity, it doesn't matter the point is it must be done geometrically. The actual flow rate doesn't matter, only the distribution.

As for the split.. I simply mean if you split a wire for example into two, and then those two into two, you get four, and then 8, and so on. Simple geometric progression there. But how can you link that to the other points when the distance to any point from an outer point varies by distance.

 

[mfb]

There is no well-defined "flow at a point", neither in electricity nor for water. You can consider flow through a surface, or along a pipe/wire.

As for the split.. I simply mean if you split a wire for example into two, and then those two into two, you get four, and then 8, and so on. Simple geometric progression there. But how can you link that to the other points when the distance to any point from an outer point varies by distance.

Distance does not matter for electricity.

You can split a line into 3 parts, and split all 3 parts into 3 parts each afterwards. I don't see the issue.

 

[pmacias]

I would approach this problem by first defining the parameters and variables involved. In this case, the parameters would be the 3x3 grid and the equal flow distribution requirement. The variables would be the flow rate and the paths from the outer points to the inner points.

Next, I would analyze the problem using mathematical models and equations. One possible approach could be to use a system of linear equations, where the total flow into each inner point is equal to the total flow out. This would ensure equal flow distribution to each point.

However, as you mentioned, the paths to the source would be different for a 3x3 grid compared to a 2x4 grid. This could potentially affect the flow rate and make it difficult to achieve equal distribution. In this case, you may need to consider other factors such as the resistance of each path and adjust the flow rate accordingly.

Another approach could be to use a network flow algorithm, which is commonly used in traffic flow problems. This would involve assigning capacities to each path and optimizing the flow distribution based on the constraints of the grid.

Overall, there are multiple ways to approach this problem and it would require further analysis and experimentation to determine the most effective solution. It certainly has potential for further exploration and could be used to model various applications, as you mentioned.