Niek has $16$ square cards that are yellow on one side and red on the other. He puts down the cards to form a $4 \times 4$-square. Some of the cards show their yellow side and some show their red side. For a colour pattern he calculates the monochromaticity as follows. For every pair of adjacent cards that share a side he counts $+1$ or $-1$ according to the following rule: $+1$ if the adjacent cards show the same colour, and $-1$ if the adjacent cards show different colours. Adding this all together gives the monochromaticity (which might be negative). For example, if he lays down the cards as below, there are $15$ pairs of adjacent cards showing the same colour, and $9$ such pairs showing different colours. [asy][asy] unitsize(1 cm); int i; fill(shift((0,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((0,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((2,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((0,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); for (i = 0; i <= 4; ++i) { draw((i,0)--(i,4)); draw((0,i)--(4,i)); } [/asy][/asy] The monochromaticity of this pattern is thus $15 \cdot (+1) + 9 \cdot (-1) = 6$. Niek investigates all possible colour patterns and makes a list of all possible numbers that appear at least once as a value of the monochromaticity. That is, Niek makes a list with all numbers such that there exists a colour pattern that has this number as its monochromaticity. (a) What are the three largest numbers on his list? (Explain your answer. If your answer is, for example, $ 12$, $9$ and $6$, then you have to show that these numbers do in fact appear on the list by giving a colouring for each of these numbers, and furthermore prove that the numbers $7$, $ 8$, $10$, $11$ and all numbers bigger than $ 12$ do not appear.) (b) What are the three smallest (most negative) numbers on his list? (c) What is the smallest positive number (so, greater than $0$) on his list?