Is it possible to place the numbers $1,2,\ldots,2014$ around a circle so that any number is divisible by the (positive) difference of its two neighbors?
This problem was given to me by my neighbor's daughter, and I couldn't figure it out. I'm guessing that such a configuration is impossible, but I can't see the (presumably small) trick leading to a proof. Help is much appreciated. Thanks!
Note that there are $1007$ odd numbers. Note that we cannot have an odd number surrounded by two evens, or surrounded by two odds. Thus to fufill the given conditions, odd numbers must be in groups of two, which is clearly impossible.