## Puzzle Archive

The answer to this month’s puzzle is that there are **2^n – 1** such triangles.

While the full proof has a lot of tricky details involving Pick’s theorem and modular arithmetic, here is a sketch of the main ideas:

Take an acute triangle XYZ and a vertex X, and assume X is closer to Y than to Z. You can generate an acute triangle X’YZ with double the area by extending the side YX to X’ where the distance of X’ to Y is twice the distance of X to Y. We say that triangle XYZ “generates” triangle X’YZ. See Figure 1.

Note that if YX = XZ (the triangle is isosceles), then X’YZ is a right triangle and we should not count it.

Figure 2 shows the set of valid triangles for n=1, 2, 3 along with an example triangle from n=4. An arrow from triangle A to triangle B means that triangle A generates triangle B

It turns out that the number of acute isosceles triangles is simply n. Each isosceles triangle at level n generates one scalene triangle at level n+1. Each Scalene triangle at level n generates 3 triangles at level n+1.

If the midpoints on each side of a triangle are each lattice points, then the triangle behaves differently (see the two orange triangles in Figure 2). If it is isosceles, it is generated once instead of zero times, and if it is scalene, it is generated three times instead of once. Note it is easy to count the number of such triangles because connecting the midpoints gives you a triangle at level n-2 (there is exactly 1 scalene triangle at n=2, so there is exactly once scalene triangle with lattice midpoints at n=4)

From here you can build a recursive formula for the number of triangles at each n, and you find that the total number is 2^n-1.

Congratulations to everyone who solved this month’s puzzle!

Correct Submissions

Hutama

Heidi Stockton

Calvin Pozderac

Diego quarati

Guillermo Wildschut

Student person

Rd

Corwin

Krishal

Stephen Cappella

Jordan Rinder

Pip

Sam Gentle

Willem Hoek

Sriharsha Bangaru

Megaserg

Allen Zhu

Paf

Brian Whitney

Kevin Gu

Rjtobin

Jpgauthier

Just Alpaca

Jean-Pierre Bianchi

Brian Albert

Karthick Gururaj

Diljit Singh

GCY

Shwetark

Jerry Huang

Jane R

Varqa Kalantar

Ben Mills

Guy Aglionby

Idio

Monotof

Valerii G

Kristof S

Brian Phillips

Nkotler2014

Liam Connell

Andrew Hwang

Nikos Bosse

Walter Mitty

Robert L

Sergei

David Hashe

Vinay Kameswaran, Shawn Ng, Dave Cox

Peasley

Jeffrey Shi NYU 2021

Dylan Peifer

SR

Takaki