Est. 1998 — 27 years of woodworking knowledge
Friday Puzzler -- Squares
Alex Y
A two-part puzzle today.
I. I have sixteen points in an evenly-spaced 4x4 square grid pattern How many distinct squares can be drawn using points on this grid to define the corners of each square?
II. I have another square grid of points. The number of squares that can be drawn using these points as the corners is exactly four times the number of points in the grid. How many points are there in the grid?
Re: Friday Puzzler -- Squares
Larry Barrett
I am a little confused by the problem. You say "sixteen points in an evenly-spaced 4x4 square grid pattern". I think 4 points define a single square, 9 points define a 2x2 square, 16 points define a 3x3 square, and therefore 25 points define a 4x4 square.
Anyway, assuming the question is how many square can be drawn within the 4x4 square, the answer is 16 1x1 squares, 9 2x2 squares, 4 3x3 squares, and 1 4x4 squares, or 30 squares. for the general case of a kxk square, defined by (k+1)^2 points, the answer is the sum of (1^2 +2^2 + 3^2 + ... (k)^2).
The second question at first seemed to be straightforward: just use the general formula (and a spreadsheet) to find k such that the # squares = 4*(k+1)^2.
I don't think there is a solution to this, but note that for this problem you don't say "evenly spaced" square grid - which I hope is a clue for progress.
Christmas shopping is done so have a little time to think.
Hope you and all WoodCentralians have a nice holiday - thanks for the puzzles.
Clarification
Alex Y
Larry pointed out a problem in my wording. I said 4x4, but did not mean that as dimensions. I meant a pattern of 4 rows of 4 dots per row. Nothing tricky about the dots here. Using Cartesian coordinates, they are (0,0), (0,1), (0,2), (0,3), (1,0), ..., (3,2), and (3,3)
So, set's get part 1 first!
For the second part, yes, the dots in the second part are similarly evenly spaced, in a square grid pattern.
Re: Clarification
Larry Barrett
And then there are diagonals! I completely missed the squares that are on diagonal lines, ie with sides that are at an angle to the grid. For the 4x4 square (4 points x 4 points) I think there are 20 squares (14 squares with sides parallel to the grid and 6 squares with sides at an angle to the grid).
Once you get beyond the 4x4 grid, the diagonal squares are numerous and confusing to draw. I have not found the answer to the second part yet, so may not even have the first part correct.
Correct on part 1
Alex Y
There are 9 squares of area 1, 4 squares of area 2, 4 squares of area 4, 2 or area 5, and one of area 9.
Re:Part 2
Larry Barrett
and if I counted correctly, there are 196 squares in a 7x7 dots grid.
Correct!
Alex Y
Merry Christmas to all. Puzzler will be off the rest of the year.