A fairly common Sudoku variant is so called "Thermometer" Sudoku. In this variant arrows with a rounded end (which look like a thermometer) are added to the grid, with the rule that the numbers on the thermometer must be strictly decreasing in the direction the arrow points. Thermometers can only move between squares that are directly adjacent (no diagonal moves or jumps).
Here is an example board:
The neat thing is that with thermometer's added to the mix it is no longer required for the puzzle to start with any numbers on the board.
The question here is with a single thermometer and no additional clues what is the largest portion of the puzzle that can be filled out. Your score is thus a number out of 81.
The board should have at least 1 solution with a perfect score having exactly one solution (fills all 81 squares). (Although it should be a fairly simple corollary of this paper that no such thermometer exists)