One law rules every line. Put the sheet back in rule.
How to play
Every row and every column of the sheet is a line, and one hidden law is meant to hold for all of them.
Under the sheet is the count: how many lines break the law, never which. Touch a cell to ink it or clear it - that changes its row and its column at once, and the count moves.
The laws it could be are the keys beside the sheet. A law that could not have given the counts you have seen strikes itself out; naming the right one is free, naming a wrong one costs a touch. The day is won when no line breaks the law.