Kudos to Dave Gale and chris maths for their great posts about introductory lessons that inspired the questions I pose below. At this point, I don’t have an answer to the query, but I welcome any insights and particularly any other spin-off ideas you may have.
If you have a standard sheet of square grid paper whose dots are exactly 1 unit apart and I ask you to draw a square of area 1, a square of area 4, and a square of area 9, you would probably quickly respond with the following.
Then I ask you to draw a square of area … [deliberate pause] … many immediately begin to think of continuing the pattern to area 16, but instead I ask for a square of area 10. Whether you know the Pythagorean Theorem or just how to compute the areas of squares and triangles, some experimentation hopefully will lead you to some form of the following figure which shows a square with area 10. The real twist for students here is that they need to adjust their point of view from what I’ll call horizontal squares (above) to tilted squares (below).
So, here are my questions:
1) What square areas can be created using square grid paper?
2) What areas of squares can be created more than one way?
3) Is there a largest square area that can NOT be created using square grid paper?
STOP! STOP! STOP!
Do not read any further if you want to work on these questions yourself.
What follows are my musings on these questions and some definite spoilers are included.
Perfect squares (1, 4, 9, 16, 25, 36, …) obviously can be found using increasingly larger horizontal squares. Dave’s ‘blog post gives a great start at the non-perfect, tilted squares.
Image source here.
The information he gives in the image above leads to areas of 2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, …
Merging the “tilted” list with the “horizontal list gives 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, …
The missing square areas are 3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, …
Whether this list ends depends on the solution to posed question #3.
Comparing the horizontal and tilted lists, the first area that can be found both ways is 25. I briefly thought that was an amazing find until I remembered that the smallest integral Pythagorean Triple is 3-4-5. So a tilted square whose vector position (using Dave’s language) can be expressed as [3,4] can also be created using a horizontal side of 5–the Pythagorean Theorem arises!
There may be other ways to get equivalent square areas (I’d love to hear any if you know some!), but any integral Pythagorean Triple represents a square area that can be represented at least two ways on square grid paper. There are an infinite number of such equivalences.
I don’t know the answer to this, but I think I’m close. I’ll post the problem before I finish it for the fun of letting others into the enjoyment of solving what I think is a cool pre-collegiate level math problem.
I did notice that the missing areas at the end of my discussion of question 1 seems to include a large number of multiples of 3, excluding of course, the horizontal squares with sides that have lengths that are multiples of 3. So is it possible to prove that any area that is
– a multiple of 3, but
– not a perfect square
can not be drawn on square grids?
If so, then there is no maximum area of a square that cannot be drawn using square grid paper.
If not, then the solution to this question may lie in a direction I have not conceived.
Again, I don’t know the answers to questions 1 or 3. Discussion is welcome and encouraged.