I want to complete the cross figure shown here. I would like all eleven answers to be prime numbers and none to have a leading zero. Also I don’t want any digit to appear more than once in any row or column or on either of the two long diagonals.
Given the starting values shown, how many different ways could I complete the grid?
Reveal Solution
We use 
for “1 down”, etc., and introduce notation as shown here:
for “1 down”, etc., and introduce notation as shown here:
Evidently 
and we rule out 5 of the possible six permutations as follows:
- If
or 
or 
then it quickly follows that 
and 
. This implies that 
and this clashes with either 
or 
and these cases fail. - If
then 
and there is no way to choose 
. - If
then 
and 
which are numbers that neighbour 
.
We are left with 
and the grid has been narrowed down to the summary on the right here:
To finish the job we look at cases according to 
. There are 4 possibilities:
- If
then every square is uniquely defined except 
.
So there are two possible grids here. - If
then 
and we can now consider two subcases according to
- If
then 
and 
. The grid is unique in this case. - If
then 
and 
. Four different grids here.
- If
- If
then the grid is unique except 
. - If
then the grid is uniquely defined except for 
.
There are eleven ways to complete the grid.
