Here is my guess.
The shortest is 20, and the longest is 29.999…
I could say the longest is 30, but then I am not sure the resulting shape qualifies as a triangle.

Shortest is infinity small. Longest is 29.99999999999999………

@ragingloli , That is correct. I suspect that is more than a guess.

Here is the reasoning that I used.
The average of the 3 sides is 60/3=20. If one side is less than 20, at least one other side must be greater than 20, so the longest side can’t be less than 20. if all 3 sides are 20, we count that as being the longest side being 20, and for any value over 20, we can always have the longest side greater than either of the others. so longest side >= 20.

The triangle inequality limits the length of the longest side. The sum of the other two sides must be strictly greater. If we treat the sum of the two other sides and the longest side as two values, their average is 60/2 =30. The sum of the two smalller sides must be strictly greater than 30 and the longest side must be strictly less than 30. Longest side < 30.

Putting this together we get 20 <= longest side < 30.

I’m with @gondwanalon If you have two pieces that are just slightly less than half, you put them as an inverted “V” with the piece that is left over connecting the wider end. Still a triangle.

That does not count as a triangle. The triangle inequality says that the sum of any two sides of a triangle is greater than the third. This will hold true provided that the sum of the sides of the two shortest sides is greater than the longest side.

I thought that a triangle is a shape with three sides and three corners. You did not specify what kind of triangle.

I’m with @gondwanalon again. I immediately went to an Isosceles triangle (2 sides of equal length) and it sounds like you were picturing a Scalene triangle (0 sides of equal length). But since you didn’t specify, what @gondwanalon answered initially is the answer to your question.

Additionally the triangle inequality theorem does hold true for our answer. This theorem says that the sum of any two sides is longer than the third side. In our answer if the two long sides are equal length, if you take one of the long sides and add the miniscule amount of the third side, it is still longer, by that miniscule amount, than the other long side.

The longest side cannot be infinitely small, because at least one of the other sides will be longer, so what we were calling the longest side would not be the longest side. If that does not answer your question then I need a specific example giving the lengths of the sides of a triangle such that the sum of the lengths is 60 and the longest side is not between 20 and 30.

It’s type IS a very common math problem, and yes it’s middle-school level, or basic high school level. IIRC, it or something like it is always covered in Geometry and in math aptitude tests.

Its writing needs work, though:

* The wording “A stick that is 60 inches” is too short for a math problem. You should specify that’s how long it is.
* “parts that is” -> “parts that are”
* “smallest and the largest” -> “shortest and the longest”
* ” It does help to know what is meant by average and how it can be used.” – Does it? The problem itself doesn’t mention an average.
* “The problem also requires understanding the triangle inequality.” – No it does not.
* Like many basic geometry problems, it can be solved by logic if one understands arithmetic and knows what a triangle is. And that tends to involve more real understanding than invoking formally named theorems.

More importantly, to me: Like most math story problems, it tries to claim relevance to the real world by mentioning a real thing (the stick) and them immediately betrays that premise by failing to take it seriously at all.

That is, no you can’t break a stick into a super-tiny sliver, and call that part of a triangle. At the very least, there should be a line about what the shortest piece that can be broken off the stick is, and that you can’t break the stick at an angle to create sharp ends that total longer length than the original stick, which you totally could do with a real stick.

If you’re not going to take the stick seriously, don’t undermine everyone’s integrity by mentioning it – just talk about a theoretical triangle.

Thanks for the corrections in the wording.

It may be possible to solve the problem with no understanding of averages, but it seems to me that understanding the concept makes the solution a whole lot simpler. I did not use the tem average in the question but I used it in the answer. The average of 3 things adding to 60 is 60/3=20 and the average of two things adding to 60 is 60/2=30.

You absolutely need to know that the sum of two sides of a triangle is greater than the third side. It makes no difference whether or not you have heard this described as the triangle inequality or just know it intuitively because the shortest distance between two points is a straight line.

As for using sticks, it seems a lot more fun using real objects, even in an abstract way. I think most people would understand what I intended.