How to find this minimum number on an intuitive level is understandable (see figure)

enter image description here

Let the side of the square n be given, then this number equals:

min = 4 + (n^2 - (3*floor(n/2)^2 + ceil(n/2)^2) (1)

that is, for even n , min = 4 , and for odd min = 4 + кол-во единичных квадратов

But it is not clear how to formally prove the formula (1)?

That is what this is

(n^2 - (3*floor(n/2)^2 + ceil(n/2)^2)

there is nothing like the minimum required number of unit squares.

update

as it turned out, formula 1 is not optimal in the general case (see the @Harry example)

Then the question is:

How to find the minimum number of squares besides yourself with which you can cover this square?

  • Do you need to prove the formula for your decision, or to prove that your decision is the best? In fact, the square is covered with one square, by the way :) - Harry
  • @Harry that this solution is optimal) well, it is clear that the coverage itself does not take into account, due to the triviality - ampawd
  • And it is not optimal in the general case ... - Harry
  • @Harry is why? - ampawd

1 answer 1

In general, your solution is not optimal. For proof, fortunately, a counterexample is enough - here it is:

enter image description here

On the left - your decision, on the right - a little better (I will not argue that it is optimal).

Ugh, well, I give ... Immediately it was generally possible to divide the 9x9 square into 9 3x3 squares! Okay, I’ll leave my shady place ... :)

  • 3
    mdauj, the triviality of this problem more and more evaporates - ampawd
  • It’s not for nothing that I didn’t really like geometry since childhood :) - Harry
  • one
    @Kromster right it seemed, in general, the whole question is to find the optimal formula - ampawd
  • one
    @Kromster Well, of course, if you radically change the question along the way - then all the answers to the previous one will be nothing more than comments :) You did not try the task "to find three numbers, the sum of the first two squares is equal to the third one" change to "and for the nth degree? ", and in response to the" no such "again change the question to" prove that "? ... :) Moreover, the task - judging by the fact that she is soon 100 years old, and there is no formula - as at least there is no simple solution ... - Harry
  • one
    That is why I do not answer such "muddy" questions until I receive a normal formulation from the TS. And after receiving it and answering it, I’m sending a new question and not complementing the old one. In this case, the answer given is that the initial question, that the current one is more a comment, not an answer. - Kromster