My daughter's grade 8 math homework included the following question. We were unable to find an answer, and I think we must have misinterpreted the question, as it seems way too hard.
Fold a square piece of paper along a diagonal. You made an isosceles triangle. Fold the base in half. Find the base of your new isosceles triangle. Fold it in half. Now, fold the base of this triangle in half.
Unfold the paper and describe all the different shapes you can find.
Show the relationship between the number of triangles on the paper and the number of times you folded the paper.
Write a formula to model the number of triangles created after $n$ folds.
- After $1$ fold, we have $2$ triangles, each $1/2$ the size of the square.
- After $2$ folds, we have $8$ triangles: $4$ that are $1/2$ square and $4$ that are $1/4$ square.
- After $3$ folds, we have $16$ triangles: $4$ that are $1/2$ square, $4$ that are $1/4$ square, and $8$ that are $1/8$ square.
- After $4$ folds, we have $44$ triangles: $4$ that are $1/2$ square, $8$ that are $1/4$ square, $16$ that are $1/8$ square, and $16$ that are $1/16$ square.
It's clear that there is some kind of pattern here, but it's not obvious what that pattern is. My best guess is that they intend us to include only the smallest triangle created at each step, and ignore the overlapping triangles, in which case the formula would be $2^n$ triangles for $n$ folds.
If not, how would one go about finding a formula for this? I'm fairly certain that we have counted the triangles correctly, but if not, that would be helpful information as well.