### Q&A with experts about bashing

Heyy everyone! From the title, you can see that we are going to talk about BASH. Yesss, we are going to discuss whether you really need to learn bash or not?

Let me first introduce myself, I am Pranav Choudhary, a 10th grader from Haryana(India). I like to do geo and combo the most :PP.

Oki so let's begin!
For those of you who don't know what bashing is, lemme give you a brief introduction first. Bashing is basically a technique used to solve Geometry Problems. In general, when you try a geo problem you might think of angles, similarities, and some other techniques (e.g. Inversion, spiral similarity etc). All of which are called synthetic geometry. But sometimes people use various other techniques called "bash" to solve the same problems.

Now there are different kinds of bashing techniques, e.g. - Coordinate bash, Trig Bash, Complex bash, Barycentric Coordinates. Let me give you a brief introduction to each of them.
• Coordinate Bash: You set one point as the origin, and set up $x$ and $y$ axes. Then write out the coordinates of all the required points, equations of the lines/circles involved etc. And then prove the required condition. It can get very messy if there are circles involved.
• Trig Bash: Using sine rule and cosine rule to relate lengths and angles is known as trig bashing.
• Complex Bash: You set up a unit circle, and consider that everything is placed in the argand(complex) plane. Then you write the complex coordinates of the required point and prove the desired condition.
• Bary Bash: You set up a reference triangle, and then you write out barycentric coordinates of the points involved and use them to prove the desired condition. The good thing about it is, if in a problem the vertices of the reference triangle are used heavily then things become easier
Don't worry if the above stuff sounds too vague right now, we'd learn about them in upcoming posts :D

Now you must be wondering: what's the point of this post then? So the thing is, bashing is a bit different from synthetic geo. People have different opinions about bash and it's sometimes very confusing whether you should actually learn bash or not. So I thought to make this task a bit easier by adding views of a few seniors on bashing in a Q/A format :DDD.

Q) Is it fine to bash?
"Yes, if you want to prioritize test scores over having fun. There's no rule against bashing, and if you feel like you don't like synthetic geometry anyway/are very bad at it, then you could instead learn to bash." ~ Pranjal Srivastava

"Yes. Although finding a synthetic solve is much more satisfying, there is nothing wrong with bashing." ~ Ezra Guerrero

"Yems" ~ Shourya Pandey

Q) How often do you bash?
"I don't." ~ Pranjal Srivastava

"I'd say one out of every twenty geo problems I solve involve some sort of bash." ~ Ezra Guerrero

"As you go higher, u bash less" ~ Mr. Y

Q) Is it fine if I learn bash only for contest scores PoV and in general practice I do pure synthetic like 95% of the times?
"As far as scoring in an Olympiad is concerned, I'd say it's good to have different approach options ready. Certainly, it's harmless and beneficial if you can learn some new techniques — doesn't matter if it's a bash or a new theory." ~ Mr. X

"Yes. But if you want to potentially use bashing in contest, it's probably good to have at least some practice. Bashing takes some intuition that you really only learn from bashing a bunch of problems. Probably most importantly, being able to tell how messy the computations will get given the setup, and how much messiness you're willing and able to tolerate -- this depends person-to-person, and practicing is useful not just for expanding that limit but also knowing where that limit is in the first place.
There's also other little things you pick up about how to set up a problem, by seeing them when you bash another: how to deal with a radical axis or compute the equation of the incircle in bary, how to think of arguments as directed angles in complex, and many others. And about how to perform the computation itself: using Conway's notation in bary, keeping expressions factored in complex, factoring by guessing the roots. Things which you can just read the answer to, but I think you understand them much better if you try to use them yourself. This is especially important because you usually don't get partial credit for a bash, so if you start one in-contest, you want to be confident that you'll finish it. As an anecdote, the first time I saw bashing was at MOP 2019. I solved a single-digit number of problems on the handout, and the next day I tried to complex bash ISL 2018 G2 on a test. I did not get anywhere close to successfully finishing, and in hindsight my setup was completely infeasible, but I didn't know complex well enough to see that at the time. It is definitely good to solve problems synthetically. But if you plan to use bashing as a backup in-contest, you probably don't want your first time bashing a nontrivial problem to be during the test." ~ Sanjana Das

Q) I think of both synthetic and bash in a problem, not only bash. Is it fine ?
"I think this is actually the optimal way to bash. Bashing and synthetic aren't mutually exclusive -- there's a lot of general trying to get a foothold on the problem that helps with both synthetic and bashing, and often you can simplify a bash a lot by making a few synthetic observations first. For example, choosing a reference triangle: if there's a triangle from whose perspective all the points have nicer definitions, then chances are that this is useful for thinking about the problem synthetically, but it's also useful for bashing. (Similarly, if you can find a nicer characterization of a point (or line or circle) than the one given in the problem, this would likely simplify computation if you chose to bash.)
I don't think I ever started a problem planning to bash it (except ones I did specifically to learn or practice bashing, such as from EGMO Chapter 6 and 7 or OTIS handouts). Most of the time I just started fiddling with the problem trying to make synthetic observations, and ended up getting a good bash setup out of it." ~ Sanjana Das

"Yep! There are scenarios where it's hard to imply a result synthetically which might be apparently true by "neat" bashes. So yeah, having a versatile mindset in regard to a problem is perfectly fine." ~ Mr. X

Q) What gives you more pleasure - bashing in a contest and getting 7/7, trying Synthetic in contest even if you weren't able to solve fully?
"I am happier if I solve the problem than if I don't. But also, I think bashing can sometimes be pretty satisfying. I think synthetic (olympiad) geometry is quite beautiful, although I'm not very good at it. But I really like computational geometry because of the idea that often comes up of "given this information, how can I find that other information?" and I think bashing often contains the same idea." ~ Sanjana Das

"Depends on the problem. In most cases, a synthetic approach to a problem is way more satisfying because you aren't just following a pre-defined algorithm — plus unlike bashes, these approaches don't drain you off with manipulations. However, if the problem is shouting "BASH ME", then sure go for it. Neat bashes can be more satisfying than long synthetic solutions in a few cases." ~ Mr. X

Q) According to you, why in general bashing is not liked in the math olympiad community?
"In my opinion, the anti-bashing community believes that bashing a problem is kinda(?) disrespectful towards the problem statement. The sole purpose of an Olympiad Geometry problem is to comment on a person's ability to tackle somewhat tricky points, which, however, bashes appear to neglect in most cases. Moreover, bashing a problem makes our solution-focused towards the final goal — missing out on various cool properties one might figure out synthetically. Basically, it destroys the beautiful journey of solving a problem by implications of a pre-defined algorithm." ~ Mr. X

Q) People say you need synthetic observations for bashing too. How much do you agree with it?
"I don't really bash, so I don't personally know. The general consensus seems to be that even while bashing, one should make synthetic claims in conjunction." ~ Pranjal Srivastava

"Synthetic observations definitely help a lot when bashing. Most of the time that I bash a problem I started out with some preliminary synthetic observations that facilitated the computations." ~ Ezra Guerrero

"Sometimes, yes" ~ Shourya Pandey

"Depends on how much can you compute by hand" ~ Mr. Y

"Very much. Many times, a problem might not look bash friendly, but a few synthetic observations might make it one! For instance, you were thinking of a complex bash to a prototype problem but you're supposed to find the complex number of a point $X$ which, in the problem was defined as the intersection of two circles. Going for a straightforward bash might go messier but suppose you acknowledge some nice property of $X$ by synthetic observations (like you found $X$ is the midpoint of $AB$), then the calculations can be drastically improved." ~ Mr. X

Q) Did it affect your synthetic skills?
"Yes, I think there is a minor negative impact. When I used to do trig, I was worse at synthetic. When I stopped trig, I improved. I think in general ppl start looking at problems a bit too 'structurally' once they bash. To some extent this can be helpful, but it shouldn't be overdone. I feel that while bashing people look at a problem in a slightly different way as to when trying the problem synthetically.
If you learn bash, the net result is probably going to be higher test scores, so to that extent becoming worse at synthetic geo shouldn't matter." ~ Pranjal Srivastava

"Surprisingly, I believe that it has improved my synthetic skill, especially when I don't want to carry out a 20 page calculation :laughing:. Learning several bashing techniques, although not required, is in my opinion, very helpful. My favorites are Bary, Length, Complex, Cartesian in that order." ~ Ezra Guerrero

"No, because when I used to bash I didn't just quit doing synthetic.It's like saying 'if you study topic XYZ it impacts topic ABC'. It doesn't have to. It'll only impact it if you stop doing ABC altogether." ~ Shourya Pandey

"At the peak, yes. I failed to see simple synthetic observations but keeping a balance helps. Dont overbash." ~ Mr. Y

"Well bashing occasionally depending on the problem doesn't affect it. However, getting obsessed with bashing can certainly make you more biased towards it which will make bashing your first preference whenever you visit a problem." ~ Mr. X

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Here are views of a few people on bashing in general which are not bounded by any such question :-

Vivian Loh : I rarely ever do 100% non-bash or 100% bash. I usually do the first like 60-85% of the question without bash and then sometimes I have to do a tiny bit of trig, coordbash, evaluating ratios, etc to finish off the problem. I mean I do quite a bit of 100% non-bash, but most of the time some of each.
Like for example if I just need to prove something is harmonic, I convert it into cross ratios so its just like 2 fractions that must be equivalent and then i might use something algebraic to show that they are equivalent. That's just an example, but I basically never use bary or complex.

Atul Shatavart Nadig : There was a point of time when I used to bash a lot and it severely impacted my synthetic, like every problem i looked at, I only looked at things like "what do i take as unit circle to complex this", "can i just spam trig and finish at this point" and honestly just wasn't able to do synthetic anymore. Also I've seen many peopll who very much dislike geo and when i ask them why, they say "you can just bash it", which i find quite silly, if you're doing bash as just a way to avoid preparing for Olympiad geo, fine, but if you actually want to appreciate geometry and have fun with it, then stop bashing and start doing synthetic.
Essentially my point is, if you only care about geo as far as "I need to be able to solve geo on a test even if its no fun" then go ahead and learn bash and bash every geo and die when there's a non-bashable geo and people like me will laugh at you then, but geometry is different from other parts of oly math. its something that has no relevance in higher math, true. but do geo anyway because its fun and super elegant. i personally find geo one of the most satisfying things to solve in math, please dont convert such a beautiful thing into pure ugly algebra which no one cares about #nobash.

My own views : I still haven't learned complex bash quite nicely, and I don't think I actually will. I learned bary bashing a few weeks ago. And I think it is quite useful from a contests point of view, like many questions are easy by bary (here easy means - take a decent amount of time, not too much, and sometimes less than synthetic if you ain't that good at synthetic and are also not that messy). But the thing is I think I started seeing the problems with - ok this can be the reference triangle and then...... I do solve the problem with bary quite quickly most of the time. Tho I try with synthetic afterward too, the problem is I am no longer that patient with synthetic - coz I feel "mehh! I do have a solution, let's just see a hint" which is surely hampering my synthetic skills. I don't dislike bash but I personally thought to not bash anymore unless required very much. And imo bash is generally a good weapon for contests, so there's no harm in learning it unless you are over-bashing problems.

I hope this post helped you decide whether you should learn and use bash or not :D. And just remember whatever you choose is absolutely fine! And it's your own decision, which means you feel that's correct and it surely is!

At last, thanks a lot to all my friends and seniors, Pranjal, Shourya, Sanjana, Atul, Vivian, Ezra, Mr. X, Mr. Y (oops they asked me to keep their identities private so I just used letters for them xDDD) for taking out time to write such lovely answers!

Have a great day :P
Pranav

1. wonderful blog

2. very cool post pranav bhaiya :)

3. Loved your explanation and the whole setup of the core concept and whereabouts of bashing......... Cheerio

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