Let's start by filling out the gaps in the grid with letters to make it easier to describe:
.----.---.---.---.
| 5 | y | 3 | x |
:----+---+---+---:
| a | | | c |
:----+---+---+---:
| b | | | d |
:----+---+---+---:
| 12 | p | q | 6 |
'----'---'---'---'
The sum of all the numbers in the grid must be 4n - (5 + 6 + 12 + x)
, where n
is the sum of each edge, and x is the missing corner
number. This is because the numbers in the corners are counted twice
each when summing the edges. As we know the grid is filled by the
numbers 1 to 12 inclusive, we know that this is equal to 78: 78 = 4n - (5 + 6 + 12 + x)
.
We now need to find x
. Rearranging, we get 78 + (5 + 6 + 12 + x) = 4n
, so 101 + x
must be divisible by 4. As 3
has already been used
in the grid, we are left with the possibilities of 7
or 11
.
If x = 7
, n = 27
, so y
in our diagram must be equal to 8.
Looking now at the left column, the only remaining pair of numbers
that sum to 10
for a
and b
are 1
and 9
. Moving to the right
column, c
and d
need to sum to 14. The only pair of numbers that
fulfils this requirement is 4
and 10
. Finally, looking at p
and
q
on the bottom row, we need these two numbers to sum to 9, however
the only numbers remaining are 2
and 11
. We can therefore exclude
the possibility that x = 7
.
We now know that x
must equal 11
. This means that n = 28
, so y = 9
. Looking at the bottom row, we need p + q = 10
. The only free pair of numbers which fulfils this is 2
and 8
. Moving on to the
left-hand column, we require a + b = 11
. The remaining possibilities
are 1
and 10
, or 4
and 7
. Finally, we look at the right-hand
column - we again need c + d = 11
. The only solution remaining is
the pair of numbers that weren't used to fill a
and b
.
The number of correct solutions, therefore, assuming we fill the numbers in in
that order, is the number of choices for p
, multiplied by the number
of choices for a
, multiplied by the number of choices for c
. This
is equal to 2 x 4 x 2 = 16
.