What does 'up to' mean in mathematics?

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The phrase "up to" in mathematics indicates that while multiple solutions may exist, they are considered equivalent under certain transformations, such as isomorphisms or reorderings. For example, in the context of Rubik's Cube, there are 24 distinct configurations, but they can be viewed as one solution when accounting for rotations. This concept emphasizes the idea of equivalence classes, where different representations do not alter the fundamental nature of the solution. Understanding "up to" helps clarify discussions about uniqueness and similarity in mathematical structures. Overall, it highlights the importance of context in determining how solutions relate to one another.
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In the context such as 'up to isomorphism' or 'up to reordering', I've seen it a lot but never understood.

Thanks!
 
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"The solution is unique up to isomorphisms" means that there are several solutions, but they are all isomorphic.

Think of Rubic's cube. There are really 24 solutions (6 different faces can point up, and then you have 4 different face that can point forward). But essentially ther are all the same. You could say "there is one solution to Rubic's cube, up to rotations of the entire cube.".
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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