No! Not a firing squad nor the need for a continuous line to cross all line segments just once!
To each vertex labeled A to L apply a different value from 1 to 12. Let V, W, X, Y and Z be the sums of their respective surrounding vertices.
Provide at least one example where V=W=X=Y=Z, or offer a reason why this, like the continuous line, is impossible.
I agree that there are no assignments to A .. L which will yield the required equality of the five sums.
Before reaching that, I was (and am) not sure whether the term vertex/vertices here refers only to the CORNERS of rectangles, or to all of the label points. Sums for V and X are clear, but the sum for W might or might not include G; similarly, the sum for Y might or might not include F; and for Z might of might not include H.
I almost stopped since the specs seemed ambiguous on this point, but taking a brute force approach I tried it both ways (i.e. including or excluding the non-corner points in the sums for W Y and Z. Machts nicht -- no solution either way.
I agree that problem statement was odd, when asking for a "reason why" there was no solution if that were the case. I think Charlie also used what appears to be an exhaustive search, but that is a "reason" to be avoided if possible (and pehaps Charlie offers one that I missed).
My guess would be that a reason might be found by considering that 6 of A..L will be ODD and the other 6 EVEN -- and that somehow that can be used to argue that not all sums can be even, or all odd (in which case, QED), and a fort. that they cannot all be a single value. The computer program cold do the brute search in under a second, and I decided to let it go. I'll take another look at Charlie's gambit.
vertices, etc.t
