Problem 16. David wants to place the numbers 1 to 8 in the 8 cells of the diagram, with one number in each cell. He wants the cells that contain two consecutive numbers not to share a side or a vertex. Which numbers can David put in the cell marked X? So let's start filling in this diagram. And we'll start with filling in the cells that we have the most information about. So if we look at these middle two cells, we see that each of them have six neighbors that they share a side or a vertex with. The top one of the middle two cells is neighbors with every cell except for the one marked X. And the one right below it is neighbors with every cell except for the cell on the very top here. So this means that whatever number we put in the middle cells, its consecutive numbers can't go in any of those neighbor cells. So there's only one place where we can put a consecutive number. Whatever number goes here, its consecutive number has to be in the cell marked X. Whatever goes here, its consecutive number has to be in the top cell. This means that these central numbers can only be consecutive with one number. So they have to be 1 and 8, because every other number has two consecutive numbers, the one below it and the one above it. There's no way to know which order they have to be in. Either order will work. But if we put the 1 in the top cell, then the consecutive number to the 1 or the 2 is going to have to go in the X. And if we put the 8 in the top of the middle cells, then we'll have to put the 7 in the cell marked X. So 2 and 7 are going to go in this cell and this cell. And they have to go in those two cells, but either could go in either one. So the answer is B. And here, this is one fully filled example that we can do just to check that this is really possible. But we already know that the only two numbers that could go in those cells are 2 and 7.

number placement

consecutive numbers

diagram puzzle

central cells

cell X