While reading a past paper, I found this problem:
Write the positive integers 2-11 (2 and 11 included) into ten little squares shown in the picture below, so that each square has one number and the numbers in these squares are all different. If the sum of the numbers in any of the 2x2 grids are all equal to K, what is the largest possible value for K?
In case you didn’t quite understand the 2x2 grid part, there are three of these 2x2 grids in the image: in the top left, middle, and bottom right. The overlap each other by one square.
I think I’ve seen lots of these types of questions in past papers, but I just can’t solve them. Is there a concept that I haven’t grasped on that could help me with questions like this? Note that this is a test question, so I’d probably only get 5 minutes or less to work on a question like this.
Hint: If you add up all the small squares you get (what?). If you add up the three $2 \times 2$ squares you get $3K$. The difference is that two squares get counted twice in the $3K$ sum, so to make $K$ large those should be as large as possible, but there is a problem with that.