Rounding Triangles

 In order to distract myself from thinking about the disastrous US election I think I will post a mathematical problem/puzzle.

Consider an arbitrary triangle with angles P, Q and R (measured in degrees). Round these angles to their nearest integer values P', Q' and R'.

List the values that S=P'+Q'+R' can take.

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Solution:

The most that S can differ from P+Q+R=180, in absolute value, is 1.5 degrees, but as S and 180 are both integers S can only differ from 180 by -1, 0 or +1 degrees. Now trial and error shows that there are triangles for which S takes each of these values (60+0.3, 60+0.3 , 60-0.6), (60, 60, 60), (60-0.3, 60-0.3, 60+0.6)

For reference here is a histogram of 180-S for 1000 random triangles:

Diagrams in Mathematical "Proofs"

 

While trying to help answer a question on Math Help Forum (MHF) (1), I suggested that the questioner draw a diagram of the data provided in the problem statement. The question was about a circle with some information given about where it touched the y-axis, and one of the points where it cut the x-axis. (My diagram is shown in ref (1), but it could be improved by some additional constructed lines...)

The thing is, that the poster did not think of this as the first step looks to me like a result of a teaching practice discouraging the use of diagrams. If so this could be a result of the enduring influence of some of the ideas of the Bourbaki group (2). To quote the Wikipedia article: "Most of the final drafts of Bourbaki's Éléments carefully avoided using illustrations, favouring a formal presentation based only in text and formulas".

This leads to the absurdity of teaching elementary geometry "proofs" with the "two column proof" style (even though even here the first step is usually sketch the givens in a diagram):
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Example (borrowed from (3))
Proof for: If two angles are supplementary to the same angle, then they are congruent to each other.
Statements Reasons
1. ∠𝐴 and ∠𝐵 are supplementary to ∠𝐶 Given
2. 𝑚∠𝐴 + 𝑚∠𝐶 = 180 degrees Definition of Supplementary Angles
3. 𝑚∠𝐵 + 𝑚∠𝐶 = 180 degrees Definition of Supplem entary Angles
4. 𝑚∠𝐴 = 180 degrees – 𝑚∠𝐶 Algebraic Manipulation
5. 𝑚∠𝐵 = 180 degrees – 𝑚∠𝐶 Algebraic Manipulation
6. 𝑚∠𝐴 = 𝑚 ∠𝐵 Transitive Property of Equality

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The use of diagrams was a point of disagreement I had with the Prof, when I was a research student. Back then his word was law, but I still disagreed in private and thought that Bourbaki sought to deprive mathematics of meaning. The point of a proof is to gain understanding of why a mathematical statement is true (or not). Maybe later a rigorous full proof may be necessary, but surely we can leave that to the machines.

I'm not sure Bourbaki carries the weight it did 50 years ago any longer, but I still find I have to tell/ask students posting questions on MHF to draw a picture.

(1) https://mathhelpforum.com/t/tangency-point-please-help.310125/

(2) https://en.wikipedia.org/wiki/Nicolas_Bourbaki

(3) https://www.effortlessmath.com/math-topics/mastering-two-column-proofs/