I think this one is intuitive...
 

You have 10 Forks and 10 Knives.

Can you arrange them in a way that there is no group of 10 consecutive utensils that contains exactly 5 of each?

[spoiler]No. Consider S_n to be the nth sum of ten consecutive utensils for some ordering, where a knife is given a value of zero and a fork is given a value of one. We are looking for an ordering that does not yield a value of S_n=5 for any n. There are two possibilities for S_0: either S_0<5, or S_0>5. In either case, at some point we have to cross the S_n=5 threshold for some value of n (we run out of knives or forks to stay on the same side). As going from S_n to S_n+1 changes the sum by at most +-one, at some point we will have S_n=5.

It is intuitive, but I couldn't figure out how to write it succinctly.[/spoiler]

[SPOILER]
Cannot be done.

As you shift the frame one position, the number of spoons changes by by 0 or -1 from the trailing edge, and 0 or +1 from the leading edge. The net change is -1, 0, or +1. Hence as you shift from frame to frame, the count of spoons changes to an adjacent integer.

The first frame must have a shortage of spoons or forks - without loss of generality, spoons. If no frame is to have exactly 5 spoons, then every frame must have less than 5 spoons.

The first and last frames have less than 5 spoons each, and together they have all the spoons, so is every frame has less than 5 spoons, then there are less than 10 total spoons.[/SPOILER]

There is no spoon :P