1/n. Slow-moving thread on a series of related, relaxing puzzles. They all involve this game:

Label all vertices and edges of a graph. Use each number 1,2,3,... exactly once (don’t miss any). At each vertex the sum of its label and its incident edge labels must be a constant, h. 2/n The labeling is called “total” because vertices AND edges are labeled.

It’s called “vertex magic” because we form a sum at each vertex, and—magically—get the same result (called the magic constant) each time.

The 2 examples suggest questions, eg: which constants are ok&why?

(1/11) I've seen this posted a lot, but usually not with many comments. This thread has:

1. A version for teachers (ready to go for the class)

More importantly:

2. A view that promotes a richer, more human view towards algebra.
nytimes.com/2020/02/05/sci… (2/11) As a disclaimer, I didn't read the article carefully. I do what I normally do, look just enough until I see an idea, and then go and recreate a version. This is how I (and probably most) mathematicians operate.

Therefore, this may look nothing like what's in the article.

The 1/7 (of area) triangle theorem.

How knowing *about* Linear Algebra makes this an exciting and doable exercise.

A problem exploration thread (1/14). (2/14) The Problem:

Take triangle ABC. Let A’ be the point in BC that’s twice as far from C as from B.

Similarly, B’ ... on CA twice as far from A as C

C’....on AB 2ce as far from B as A

Connect AA’, BB’, CC’. Prove the middle triangle has 1/7 the area of the original, ABC.

The 1/7 (of area) triangle theorem.

How knowing *about* Linear Algebra makes this an exciting and doable exercise.

A problem exploration thread (1/14). (2/14) The Problem:

Take triangle ABC. Let A’ be the point in BC that’s twice as far from C as from B.

Similarly, B’ ... on CA twice as far from A as C

C’....on AB 2ce as far from B as A

Connect AA’, BB’, CC’. Prove the middle triangle has 1/7 the area of the original, ABC.

1/n I believe very strongly in learning guessing. But how do we teach this in the classroom?

Here’s a small example, which happened as part of a precalculus review for calc I students a couple of weeks ago:

https://twitter.com/fawnpnguyen/status/1152280509189218304

2/n The first class, I solicited guesses for the answer to the following:
find two numbers whose sum is 5 and whose product is a maximum. 2 and 3 was suggested, followed by the question: do they have to be integers?
This led to a guess of 2.5 and 2.5.

Next question and guess: