Here is a 6-by-6 board. Every cell holds one of the values 1, 2, 3, 4 or 5. Rows are numbered 1 to 6 from the top and columns 1 to 6 from the left, and (r, c) means row r, column c. The same board as text:
c1 c2 c3 c4 c5 c6
r1 4 3 2 3 1 4
r2 2 1 4 1 5 5
r3 3 5 4 2 2 1
r4 2 2 5 1 1 3
r5 5 5 1 1 4 3
r6 4 1 3 1 1 1Choose a whole number n >= 1 and one n-omino T, then place as many copies of T on the board as you like.
- An n-omino is a connected shape made of n unit squares joined edge to edge. Each copy must lie exactly on n cells of the board.
- Copies may be rotated and reflected.
- Copies may not overlap. Cells left uncovered are fine.
- A single copy may not cover two cells that hold the same value.
- A copy scores the product of the values it covers. Your total score is the sum of the scores of all your copies.
Example. Take n = 3 and T the straight tromino (three cells in a line). The copy on (1,1), (1,2), (1,3) covers 4, 3, 2 and scores 24. The copy on (5,4), (5,5), (5,6) covers 1, 4, 3 and scores 12. The copy on (3,4), (3,5), (3,6) is illegal, because it covers two 2s. Using the two legal copies gives a total of 24 + 12 = 36.
Question. Over all choices of n, T and placements, what is the highest total score you can get?
Answer format: a single integer (the maximum total score).