, 2020

Solution:

We first note that there are $2^{6}-1=63$ possibilities for lights in total. We now count the number of duplicates we need to subtract by casework on the number of buttons lit. To do this, we do casework on the size of the minimal "bounding box" of the lights:

- If the bounding box is $1 \times 1$, the only arrangement up to translation is a solitary light, which can be translated 6 ways. This means we must subtract 5.

- If the bounding box is $2 \times 1$, there is 1 arrangement and 4 translations, so we must subtract 3.

- If the bounding box is $1 \times 2$, there is 1 arrangement and 3 translations, so we must subtract 2.

- If the bounding box is $3 \times 1$, there are 2 arrangements and 2 translations, so we must subtract 2.

- If the bounding box is $2 \times 2$, there are 2 arrangements with 2 lights, 4 with 3 lights, and 1 with 4 lights -- 7 in total. Since there are two translations, we must subtract 7.

The final answer is $63-5-3-2-2-7=44$.

Solution:

We may also count duplicates by doing casework on buttons lit:

- 1 button lit: There are 6 arrangements but all are the same, so we need to subtract 5 duplicates in this case.

- 2 buttons lit: There are 4 indistinguishable ways for the buttons to be vertically adjacent, 3 to be horizontally adjacent, 2 ways for the buttons to be diagonally adjacent for each of 2 directions of diagonals, and 2 for when the lights are in the same vertical line but not adjacent. Since we need to count each of these cases only once, the number of duplicates we need to subtract is 3 (2 vertically adjacent), 2 (2 horizontally adjacent), $2 \times 1$ (2 diagonally adjacent), and 1 (2 in same vertical line but not adjacent) for a total of 8 duplicates.

- 3 buttons lit: There are 2 indistinguishable ways for all the buttons in a column to be lit and 2 ways for the buttons to be lit in the shape of an L, given the rotation of the L. Thus, the number of duplicates we need to subtract is 1 (1 column), $1 \times 4$ (rotations of L), for a total of 5 duplicates.

- 4 buttons lit: There are 2 indistinguishable ways for the lights to be arranged in a square (and no other duplicates), so we need to subtract 1 duplicate in this case.

- When there are 5 or 6 buttons lit, all of the arrangements of lights are distinct, so we do not subtract any duplicates for these cases.

Thus, the total number of arrangements is $64-(1+5+8+5+1)=44$.