# My head can't handle it

Hi All,

I wonder if you can help me with a little puzzle I'm working on. I'll include a picture for clarity. I suspect that you mathsy types will be able to tell me straight away if this is possible or not, but I just can see the obvious...

I have a fine square grid of numbers (blue in the picture), but I do not know the values of the numbers. I also have, overlaid (red in the picture), a more coarse-grained grid of numbers which are the averages of the four numbers they lay ontop of.
I then have another identical large grid (green in the pic) which is displaced by (+1,+1) small grid points. Its values are also the averages of the blue values it obscures.

My question....
If I know all the red and green values (averages of underlying data set), can I work out the blue values? If it can be done, it will no doubt result in a huge string of simultaneous equations. If somebody could start me off by showing how a small section of this could be done, I'd be very grateful. Or if it's impossible, could somebody tell me why?
Would it help to have more large grids off-set by other amounts? The grids can extend a long way, and I'm not fussed about knowing ALL the blue numbers - if I can work my way out from the middle, I'll go as far as I need, then forget about the perifery.

Thanks for any help,
Phil

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simplify

I cannot answer your question but why don't you try that with a minimal grid ( is that 5x5) i'm not sure , that will limit the number of equations you will have to consider -- off the top of my head the number of equations must match the numer of unknowns do your two grids provide that ?? Ray

matt grime