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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.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.
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.
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.
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.
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.
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.