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if my invention includes various solutions to solving the same problem such that all solutions are independent of each other, how should I claim such an invention?

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These do sound like separate inventions. Patents are given for specific solutions, not to results. Three solutions to the same problem that are technically independent are not a single invention just because they address the same problem.

If you have already filed a provisional application that discloses more than one invention it is not a problem. Many non-provisionals can get priority from a single provisional and also a single non-provisional can get the benefit of more than one provisional.

  • I agree, but all solutions are disclosed in a single provisional application. What to do in such a case? – PatentPassion Sep 4 at 17:58
  • see edited answer – George White Sep 4 at 18:06
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In the US, the basic fee for a non-provisional patent application allows you to have 3 independent claims, and further independent claims can be included by paying additional fees. You can claim each solution as an independent claim, but the Examiner may force you to restrict the application to one or some of these solutions and move the other one or ones to different non-provisionals as divisional applications.

Concerning European patent applications usually you may only have one independent claim of each category (methods and products), but R.43(2)(c) EPC allows you to have more than one independent claim per category for "alternative solutions to a particular problem, where it is inappropriate to cover these alternatives by a single claim." It is inappropriate to cover several alternatives in one single claim when the claimed subject-matter becomes unclear or not concise due to e.g. OR clauses that hinder the comprehension of the claim. However, this provision can only be used when the alternatives relate to a same inventive concept, otherwise there is no unity of invention and, as mentioned before with reference to the US application, you will be forced to restrict the application to one or some of the alternatives and move the others to one or more divisional applications.

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Is this filing in the US, EPO or some other place?
Background:
Practice varies from place to place, but can be broadly bunched together under USPTO (Restriction) and PCT (Lack of Unity). In a majority of cases, it will end up with the same decision by all the patent office, but in some, it will not. I have seen several cases where the USPTO required the Applicant to divide the application into multiple applications (Restriction- not directed to the "same species"), but other jurisdictions have not required such division. There are some other peculiarities too: the USPTO often doesn't like Apparatus and Method claims in the same application, but the PCT generally allows it (and in fact most applications have separate method and apparatus independent claims in the same application).

if my invention includes various solutions to solving the same problem such that all solutions are independent of each other, how should I claim such an invention?

In case you are drafting your claims so that the prior art or the problem to be solved is the first part of the claims, and the solution is the second part (the wherein part): If the wherein parts are different, then they are generally directed to different inventions. However, this is not a black and white rule in the USPTO: if they fall in the same narrow classification mark, and there is no additional search burden, then all the inventions (solutions) can be in the same application. This is assuming that identification of the problem in itself is not the invention- the problem is known in the art, while the solutions you're providing are new.

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