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Here's a story to make sure you understand what I'd like to ask.

The problem was that he looked a little suspicious from the moment I laid eyes on him. I checked the bills he handed over. Many of them were brend new, but the ink they were printed with didn't seem quite right. I put them in the machine to check whether the bills were valid ones.

I'd like to know the subtle difference to make sure I use right expression in a situation.

Machine checking whether the bills were valid ones.

I think using 'checking whether' is more likely to refer to a general situation, not just confined to specific situation like above. It sounds like the machine checked whether any bills were real in the past and now is also used to check and will be used to check.

Machine that could check whether the bills were valid bills.

I'm not sure I'm right, but it sounds better in this situation becasue it is like refering to specific situation by using past tense 'could', and also the situation happened in the specific past. So I think using 'that could' is the best choice.

Machine to check whether the bills were valid ones.

I think it is the same as 'checking'. Using 'to check' is like refering to general thing. So If I want to use 'checking' or 'to check', more general sentence have to be followed like 'bills is real ones' using present tense, rather than past tense like 'The maching checking whether bills are real ones.'

I suppose you could ask me why I'm doing this because I have no idea why I think this way, BUT I'd like to know the difference.

Plus, the reason I think 'to check' sounds ackward is because we were talking about an incident happened in the past, but suddenly general phrasing was brought up using 'to check', rather than 'that could check' which is involving past tense, so goes well together with phrasing what happened in the past.

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Putting bills in the machine "to check...." means that he put them in the machine FOR THE PURPOSE OF checking them, or AS A WAY to check them.

There is no better or more concise way to express this.

Sometimes people will say "he {did X} IN ORDER TO {do Y}", but that construction is now considered verbose.

All your wonderings about past or future tense, or about continuous or repeated action, are totally beside the point and fruitless. (but they do demonstrate your confusion, so it's good you mentioned them). There is only one point in time and one incident mentioned in the sentence, and it was in the past. So the sentence uses simple past ("put" is both the present and past of "put". The context clarifies that it is meant as past tense.) Using the infinitive form "to check" does not change that.

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