You may be familiar with Sudoku, a combinatorial number placement puzzle in which you fill in a 9×9 grid with digits such that each row, column, and 3×3 square contains all of the digits 1-9 exactly once.
You come across a Sudoku puzzle in which the initially populated numbers appear to be legal (no row, column, or square had the same number twice), but it is clear that trying to solve the puzzle would lead to an impossible arrangement of numbers – an unsolvable puzzle not because there is not enough information, but because it forces you into a contradictory/rule-breaking result. What is the smallest possible sum of the initial numbers?
The general idea is to use the smallest possible numbers to force a contradiction. So you would fill in 1’s such that a 1 must appear in a certain square, and then have a 2 appear in that square instead. It turns out you can do so with four 1’s and a 2, for a sum of 6. Here is an example: