This is another wonderful puzzle from A Moscow Math Circle, Week by week problem Sets by Sergey Dorichenko.
There are two parts to the problem:
(a) Place seven stars in a 4×4 grid so that, no matter which two rows and which two columns are erased, at least one star will remain.
(b) Prove that if six stars are placed in a 4×4 grid, one can always erase all of them by erasing two rows and two columns.
As always, please send your answers as comments within the blog (preferred), or send an e-mail to firstname.lastname@example.org. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.
Happy Eid to everyone!
On part 2,
When 6 stars are placed in 4 rows, either at least one row has at least 3 stars or at least two has two stars each. So you can always find 2 rows with 4 stars. Pick these two rows and remove them. Now 2 stars remain and we can remove two columns. So, we can always remove all stars by removing two rows and two columns.