How important is math "intuition"

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Dream9

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Dream9

#1 h Mar 19, 2025, 2:16 PM

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When I see problems now, they usually fall under 3 categories: easy, annoying, and cannot solve. Over time, more problems become easy, but I don't think I'm learning anything "new" so is higher level math like AMC 10 more about practice, so you know what to do when you see a problem? Of course, there's formulas for some problems but when reading a lot of solutions I didn't see many weird formulas being used and it was just the way to solve the problem was "odd".

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Amazingatmath.com

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Amazingatmath.com

#2 h Wednesday at 6:45 PM

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I feel the same way sometimes.

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gauss202

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gauss202

#3 h Wednesday at 7:10 PM • 4 Y

Y by Unicode_Master03B8, jkim0656, Pengu14, NaturalSelection

Someone once explained to me that there is a difference between “a problem,” “an exercise,” and “a puzzle.”

An exercise is a problem whose solution is based on techniques that you are expected to know. It requires a straightforward application of those techniques. This is the nature of most of the problems on exams like the SAT.

A problem is one that requires intentional thought or work to identify which tools can be used to solve it. For example, consider the 2024 AMC12B Problem 22: “Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$ . What is the least possible perimeter of such a triangle?” Solving this problem involves a series of logical steps to uncover the necessary approach.

A puzzle is a problem that might have a quick or clever solution but requires strong logical reasoning or intuition about where to start or how to look for the solution. Late-numbered Olympiad problems often fall into this category. Even if you have the requisite background knowledge, actually finding the solution can still be elusive.

I find all three types of mathematical questions—exercises, problems, and puzzles—both interesting and useful. To solve problems, you should be skilled at exercises; to solve puzzles, you should be skilled at solving problems. Even if you can handle difficult puzzles in one field, you may still need to work on exercises to build your skills in another.

This post has been edited 2 times. Last edited by gauss202, Wednesday at 7:15 PM

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noahpnguyen

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noahpnguyen

#4 h Wednesday at 7:16 PM

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everything for me falls into the category annoying or cannot solve. A lot of easy things are annoying

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Rice_Farmer

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Rice_Farmer

#5 h Wednesday at 7:59 PM

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Wdym easy things are just free points

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Dream9

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Dream9

#6 h Wednesday at 10:00 PM

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Rice_Farmer wrote:

Wdym easy things are just free points

I think what noah means is problems that are task intensive like simple addition many times. Easy things get annoying due to the lack of thinking needed so, yes, they are easy points, but a lot of those problems tire most people out.

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fruitmonster97

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fruitmonster97

#7 h Wednesday at 10:03 PM

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Dream9 wrote:

When I see problems now, they usually fall under 3 categories: easy, annoying, and cannot solve. Over time, more problems become easy, but I don't think I'm learning anything "new" so is higher level math like AMC 10 more about practice, so you know what to do when you see a problem? Of course, there's formulas for some problems but when reading a lot of solutions I didn't see many weird formulas being used and it was just the way to solve the problem was "odd".

Could you give an example or two?

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BS2012

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BS2012

#8 h Wednesday at 10:04 PM

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Easy: 2025 AMC 8 P2
Easy/annoying: Find $7^{10}$ without a calculator.
Hard (to most ppl on msm): 2025 AIME I P9
Hard/annoying: 2020 AIME II P9

This post has been edited 1 time. Last edited by BS2012, Wednesday at 10:06 PM

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gauss202

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gauss202

#9 h Yesterday at 10:21 AM • 2 Y

Y by bot1132, noahpnguyen

noahpnguyen wrote:

everything for me falls into the category annoying or cannot solve. A lot of easy things are annoying

Then you are annoyed by having to figure things out for the first time. Have fun with figuring things out. There is a beautiful joy in problem solving. If you find it annoying, perhaps take up another activity that you enjoy more.

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noahpnguyen

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noahpnguyen

#10 h Yesterday at 8:02 PM

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gauss202 wrote:

noahpnguyen wrote:

everything for me falls into the category annoying or cannot solve. A lot of easy things are annoying

Then you are annoyed by having to figure things out for the first time. Have fun with figuring things out. There is a beautiful joy in problem solving. If you find it annoying, perhaps take up another activity that you enjoy more.

I like doing fun problems like clever computations, but very easy problems in prealgebra are kinda annoying

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skronkmonster

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skronkmonster

#11 h Yesterday at 8:15 PM

Y by

BS2012 wrote:

Easy: 2025 AMC 8 P2
Easy/annoying: Find $7^{10}$ without a calculator.
Hard (to most ppl on msm): 2025 AIME I P9
Hard/annoying: 2020 AIME II P9

yeah those ones where they ask you to find some super-secret trick to shortcut through a long calculation -- those can be annoying if you don't know the trick. there are also the ones you have to bash out or do a lot of hard calculations on.

at that stage (amc 10/12/aime) you're mostly looking at high-speed problem solving techniques and obscure tricks more than actual algebra content and abstract facts.

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Squidget

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Squidget

#12 h Yesterday at 8:50 PM

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I think I realized this most when I tried writing problems, it’s just so easy to make it a bashy or annoying problem to fill up space

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Andyluo

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Andyluo

#13 h Yesterday at 10:03 PM

Y by

gauss202 wrote:

Someone once explained to me that there is a difference between “a problem,” “an exercise,” and “a puzzle.”

An exercise is a problem whose solution is based on techniques that you are expected to know. It requires a straightforward application of those techniques. This is the nature of most of the problems on exams like the SAT.

A problem is one that requires intentional thought or work to identify which tools can be used to solve it. For example, consider the 2024 AMC12B Problem 22: “Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$ . What is the least possible perimeter of such a triangle?” Solving this problem involves a series of logical steps to uncover the necessary approach.

A puzzle is a problem that might have a quick or clever solution but requires strong logical reasoning or intuition about where to start or how to look for the solution. Late-numbered Olympiad problems often fall into this category. Even if you have the requisite background knowledge, actually finding the solution can still be elusive.

I find all three types of mathematical questions—exercises, problems, and puzzles—both interesting and useful. To solve problems, you should be skilled at exercises; to solve puzzles, you should be skilled at solving problems. Even if you can handle difficult puzzles in one field, you may still need to work on exercises to build your skills in another.

I can mostly agree with this, but problems, exercises, and puzzles can mean different things to different people.

For me, an AMC 8 p25 may have been a difficult puzzle in 6th grade, that would be extremely challenging for me.

But now, I can solve those kinds of questions in less than a single minute.

My bar for what a "puzzle" is keeps increasing, as now I consider them to be AIME final 5 or even easier olympiad questions.

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ohiorizzler1434

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ohiorizzler1434

#14 h Yesterday at 10:27 PM • 1 Y

Y by Andyluo

Bro! It's all about extending yourself with problems outside of your comfort zone! With repeated practice, you will develop problem solving skills and improve as a mathematician! I believe in you!

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BananaBall00

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BananaBall00

#15 h Today at 12:08 AM

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What I see in global feed:
How important is math

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skronkmonster

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skronkmonster

#16 h an hour ago

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BananaBall00 wrote:

What I see in global feed:
How important is math

Richard: spams 'globe' emoji

Squidget wrote:

I think I realized this most when I tried writing problems, it’s just so easy to make it a bashy or annoying problem to fill up space

The problem is that most of the time, the graders and/or AoPS themselves are looking for smarter solutions to writing problems. So, they'll dock you points if you use a long or unwiedly method, because it's not really what they're looking for. Sometimes you can bash it out without losing points though.

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Dream9

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Dream9

#17 h an hour ago

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BananaBall00 wrote:

What I see in global feed:
How important is math

i just noticed

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