A 19. Tiling an equilateral triangle

Consider an equilateral triangle. We wish to tile it with n congruent triangular tiles. We will call such a triangulation a CT-n tiling. Clearly there is a CT-n tiling if n = 2 (two 30-60-90 right triangles), n = 3 (three 30-120-30 isosceles triangles) or n = 4 (four equilateral triangles).

(1) Show that there is a CT-n for the following values of n: 1,2,3,4,6,8,9,12,16,18.

(2) Find 4 infinite sequences of n such that there is a CT-n for all values of n in the sequence. (The sequences may not overlap.)

Unsolved (by me) problems

(3) Find (with proof) at least one value of n for which there is no CT-n.

(4) Determine exactly which values of n yield a CT-n.

E 36. How many solutions?

(2020 Putnam Exam, problem A1)

How many positive integers N satisfy all of the following 3 conditions:

(i) N is divisible by 2020

(ii) N has at most 2020 decimal digits

(iii) The decimal digits of N are a string of consecutive ones followed by a string of consecutive zeros.

Note: This only requires pre-college level mathematics, but like most Putnam problems it is not easy.

A 18. Setting up breakout rooms

A Zoom session has N = 2^k people. There are n breakout sessions, each into two equal-sized breakout rooms (i.e., N/2 people in each) The facilitator wishes that every person in the session meets with every other person in a breakout room at least once. Show this can be done if n = k + 1.

Note: This is the simplest version. If you want more of a challenge, you could try the case where N is not a power of 2. Obviously if N is odd, the breakout rooms cannot be exactly equal in size, but there is always an efficient solution where the numbers of people in each breakout room are always within 2 of one another.