*It doesn't have to be a rectangular sofa either, it can be any shape. Here are the specifics: the whole problem is in two dimensions, the corner is a 90-degree angle, and the width of the corridor is 1.What is the largest two-dimensional area that can fit around the corner?You are usually told how long each person takes to paint a similarly-sized house, and you are asked how long it will take the two of them to paint the house when they work together.*

*It doesn't have to be a rectangular sofa either, it can be any shape. Here are the specifics: the whole problem is in two dimensions, the corner is a 90-degree angle, and the width of the corridor is 1.*

In this case, I know the "together" time, but not the individual times.

One of the pipes' times is expressed in terms of the other pipe's time, so I'll pick a variable to stand for one of these times. Since the faster pipe's time to completion is defined in terms of the second pipe's time, I'll pick a variable for the slower pipe's time, and then use this to create an expression for the faster pipe's time: Then I make the necessary assumption that the pipes' contributions are additive (which is reasonable, in this case), add the two pipes' contributions, and set this equal to the combined per-hour rate: Note: I could have picked a variable for the faster pipe, and then defined the time for the slower pipe in terms of this variable.

But no one has ever been able to prove that for certain.

It's possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1.

The largest area that can fit around a corner is called—I kid you not—the sofa constant.

Nobody knows for sure how big it is, but we have some pretty big sofas that do work, so we know it has to be at least as big as them.— but it's the technique that they want you to learn, not the applicability to "real life".The method of solution for "work" problems is not obvious, so don't feel bad if you're totally lost at the moment.If you're not sure how you'd do this, then think about it in terms of nicer numbers: If someone goes twice as fast as you, then you take twice as long as he does; if he goes three times as fast as you, then you take three times as long as him. For solutions of constraint satisfaction problems, see Constraint satisfaction problem § Resolution.There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does Since the unit for completion was "hours", I converted each time to an hourly rate; that is, I restated everything in terms of how much of the entire task could be completed per hour.To do this, I simply inverted each value for "hours to complete job": My first step is to list the times taken by each pipe to fill the pool, and how long the two pipes take together.But squares are tricky, and so far a formal proof has eluded mathematicians.The happy ending problem is so named because it led to the marriage of two mathematicians who worked on it, George Szekeres and Esther Klein.According to the inscribed square hypothesis, every closed loop (specifically every plane simple closed curve) should have an inscribed square, a square where all four corners lie somewhere on the loop.This has already been solved for a number of other shapes, such as triangles and rectangles.

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