*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.*Many of these problems are not terribly realistic — since when can two laser printers work together on printing one report?

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.

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.

In a Pythagorean triangle, and all three sides are whole numbers. But there are also three more diagonals on the three surfaces (D, E, and F) and that raises an interesting question: can there be a box where all seven of these lengths are integers?

The goal is to find a box where A, and where all seven numbers are integers. Mathematicians have tried many different possibilities and have yet to find a single one that works.For solutions of mathematical optimization problems, see Feasible solution.In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign.When seeking a solution, one or more free variables are designated as unknowns.A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true.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.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?We also have some sofas that don't work, so it has to be smaller than those. The first three are the dimensions of a box, and G is the diagonal running from one of the top corners to the opposite bottom corner.All together, we know the sofa constant has to be between 2.2195 and 2.8284. The three letters correspond to the three sides of a right triangle. Just as there are some triangles where all three sides are whole numbers, there are also some boxes where the three sides and the spatial diagonal (A, B, C, and G) are whole numbers.

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