This was a simple puzzle, a classical application of the pigeon hole principle. I got correct answers from Deepak Goyal (my brother in NY), Mahi Saraf (as always) and Pratik Poddar (as always again).
I am taking the liberty of copying the answer from Mahi.
Part 1
There are multiple answers to this one. I am producing two one from Mahi, and another one from Deepak.
From Mahi:
The plus signs represent the squares with the dots and the ones with the underscore signs represent the empty squares
_ _ _ +
_ + + _
+ _ + _
+ + _ _
From Deepak
Solution to Puzzle #186
Part 2
When we arrange 6 stars in a 4 by 4 grid, by pigeon hole principle there will be at least one row with 2 stars so we strike that. If the row we chose had exactly 2 stars then there will be one more row with 2 stars so we strike that to. Now we will be left with 2 stars and 2 strikes hence, we can cut all the stars.
If the first row we chose had 3 stars then we will be left with three stars and 3 strikes so even in this case we will be able to strike out all stars.
The last possibility could be that the first row we chose had 4 stars in it so we will have 2 stars and 3 strikes remaining.
Therefore, We can never arrange 6 stars in a 4 by 4 grid so that by striking 2 rows and 2 columns we will still be left with 1 star.
Hope you all enjoyed the puzzle!
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