Approach me!

Geometry Level 3

Classify all values of $k \in \mathbb{N}$ such that you can completely cover a square with $k$ congruent triangles, so that no triangles overlap and the sum of their area matches the covered square.

All even k and a finite number of odd

k

All even k and infinitely many odd

k

Finite number of even and odd

k

Only a finite number of even

k

All even

k

1 solution

Brian Charlesworth
Jul 19, 2015

For even $k = 2m,$ one can draw $m - 1$ vertical lines that divide the square into $m$ congruent rectangles, and then draw diagonals for each of these rectangles to produce $2m = k$ congruent triangles.

For odd $k,$ (although this may be overkill), we can apply Monsky's Theorem to see that no odd $k$ will suffice. The answer is thus all even $k.$

(Note that congruency is a stronger condition than the equal area condition necessary for Monsky's Theorem to hold, so there may be a simpler way to deal with odd $k$ that avoids using this powerful theorem.)

I'm not sure if that's an overkill. I can't figure out an easier approach, even with the much stronger condition of congruency.

Calvin Lin Staff - 5 years, 10 months ago

I also can't think of an easier solution than Monsky and I also find this a bit troublesome.. (I don't think such a problem is suited to be put on brilliant.org, cause it is either discover Monsky on your own, know it or guess, with the latter two having no educational value.

Alisa Meier - 5 years, 10 months ago

I still like the problem since it does serve as an introduction to Monsky's Theorem. One can reasonably guess that it is just the evens, and when solvers then look at the solution they can learn about the relevant theorem, thus providing educational value. And who knows, perhaps someone on Brilliant will come up with a non-Monsky's proof sometime. :)

Brian Charlesworth - 5 years, 10 months ago

Approach me!

1 solution

0 pending reports