How Many Squares Can Be Obtained By Drawing 6 Same-Sized Squares?

Question

Question is very understandable. We have 6 of the same-sized squares. By putting/using them together, how many squares can we get at max?

For instance; if we would have 2 same-sized squares, we could get 3 squares by putting them from middle and corner. 2 from our original squares and 1 from intersection.

What exactly do you need help with solving? Please share what you have tried. — Bobson Dugnutt, Apr 22 '17 at 16:35
I have tried it with 3 and 4 squares to see if I could have pattern here.
For 1 square, 1 square. For 2 squares, 3 squares. For 3 squares, 7 squares. For 4 squares, 11 squares. (not very sure about that one)

Apparently no clear pattern or I have calculated wrong.

Now I am susp about if (2^x)-1 is our answer or not. — Oğuzhan Karataş, Apr 22 '17 at 16:37
For 3 squares to start my overlapping pattern gives 8 and 4 gives 15. — Ross Millikan, Apr 22 '17 at 23:28

Ross Millikan · Accepted Answer · 2017-04-22T23:28:16.960

1

Here is a diagram that gets $41$. I don't know if it is maximal

For $n$ initial squares, the number in this pattern is $3n-3$ plus twice A002623(n-2) except for $n=1$ it is $1$, not $0$. The twice A002623 accounts for the two stairsteps on the sides, and the $3n-3$ gets the main diagonal. This gives the sequence for number of squares in this pattern as $1,3,8,15,26,41,62,89,124,167$. This is not in OEIS

edited Apr 22 '17 at 23:28

answered Apr 22 '17 at 16:45

Ross Millikan

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How Many Squares Can Be Obtained By Drawing 6 Same-Sized Squares?

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