Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. How long will it take C alone to fill the tank ?

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Let's understand this pipe filling problem. We have three pipes A, B, and C that can fill a tank. In a special scenario, pipe A runs throughout the entire process, pipe B runs for the first 10 minutes then stops, and pipe C starts 2 minutes after B stops. Each pipe fills exactly one-third of the tank. We need to find how long pipe C alone would take to fill the entire tank. Now let's set up our variables. Let r_A, r_B, and r_C be the rates at which pipes A, B, and C fill the tank respectively, measured in fraction of tank per minute. Let T_2 be the total time taken in this scenario. The key insight is that each pipe fills exactly one-third of the tank. This gives us three equations: pipe A works for T_2 minutes and fills one-third, pipe B works for 10 minutes and fills one-third, and pipe C works for T_2 minus 12 minutes and fills one-third. Let's solve for pipe B first. From our equation, pipe B's rate times 10 minutes equals one-third of the tank. Dividing both sides by 10, we get r_B equals one-thirtieth tank per minute. To find how long pipe B alone would take to fill the entire tank, we take the reciprocal of its rate. One divided by one-thirtieth equals 30 minutes. So pipe B alone takes 30 minutes to fill the tank. The problem statement has some ambiguity regarding the parameter 't', which creates contradictions in the given conditions. To solve this problem, we need to make a reasonable assumption. Since the problem asks specifically for pipe C's time and we've found that pipe B alone takes 30 minutes, the most natural assumption is that pipe B and pipe C have equal individual filling times. Therefore, we assume pipe C alone also takes 30 minutes. We can verify this assumption works: if r_C equals one-thirtieth, then the equation gives us T_2 equals 22 minutes, which is consistent with our scenario. Let's summarize our solution. We found that pipe B fills one-third of the tank in 10 minutes, giving us a rate of one-thirtieth tank per minute. This means pipe B alone would take 30 minutes to fill the entire tank. Using our reasonable assumption that pipes B and C have equal individual filling times, we conclude that pipe C alone would also take 30 minutes to fill the tank. Therefore, the answer to the question is 30 minutes.