19. Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15 . In how many ways can she do this?
A 8
B 12
C 16
D 24
E 32
Solution
C

Let the integers in the different regions be $p, q, r, s, t, u, v, w$ and $x$, as shown. First consider the positions of integers 8 and 9 . Since the product of integers in two regions with a common edge cannot exceed 15 , integers 8 and 9 can only be placed in regions with one edge in common with another region. Therefore 8 and 9 can only be written in the regions marked with $s$ and $u$ in either order and 1 can only be written in the region marked $t$.
Next consider the position of the integer 7 . This can only be in a region with a common edge to the regions with 1 and $2 \mathrm{in}$. Hence, 7 can only be placed in one of the regions marked $p, r, v$ or $x$ and then must have a 2 in the adjacent triangle. Since the integer 6 can also only be in a region with a common edge to regions containing either 1 or 2 , once the position of 7 is decided, the position of 6 is fixed as being in the square on the opposite side of the triangle to the square containing 7 .

Finally, the three remaining regions must be filled with integers 3,4 and 5 with the 3 being written in the second triangle and the other two integers being written either way round. Therefore the diagram can be filled in $2 \times 4 \times 2=16$ ways.