Spoiler for New Scientist Enigma 1751: “Rainbow square” (Follow the link to see the puzzle.)
We have R-O-Y-G-B-V across the top row. Where can the consecutive R-O-Y-B be on one of the diagonals? Not starting at the upper left corner; that would put two Os, Ys, and Bs in the second, third, and fourth columns. Not starting at the lower left corner for similar reasons (plus two Rs in the first column.) It can’t start anywhere else on the upper left/lower right diagonal, because that would put two Rs on that diagonal. It can’t start in the second row, fifth column; that puts two Ys in column 3. It can’t start in the third row, fourth column; then the B on the upper right/lower left diagonal would have to be in column 5, but there’s already a B in that column. The only place it can start, then, is in the fifth row, second column. Then the lower left square must be B, and the V-B-G-Y-O-R row must be the third row. So we have:
R O Y G B V ? ? ? ? G ? V B G Y O R ? ? O ? ? ? ? R ? ? ? ? B ? ? ? ? ?
Where can the B on the upper left/lower right diagonal go? Only in the fourth row, fourth column. Likewise, the O in that diagonal must go in the sixth row, sixth column.
R O Y G B V ? ? ? ? G ? V B G Y O R ? ? O B ? ? ? R ? ? ? ? B ? ? ? ? O
Now the G in the bottom row can only go in the second column, and then the Y in the bottom row can only go in the fifth column.
R O Y G B V ? ? ? ? G ? V B G Y O R ? ? O B ? ? ? R ? ? ? ? B G ? ? Y O
But that means the fifth row, fifth column can’t be Y, so must be V, and the second row, second column must be Y. The diagonal not containing R-O-Y-G, then, is R-Y-G-B-V-O.
There isn’t a unique solution for the remaining squares. Here’s one possibility:
R O Y G B V O Y R V G B V B G Y O R G V O B R Y Y R B O V G B G V R Y O
or in color:
(Another solution is the same but with the G and Y squares in the first and last columns interchanged. There are others.)