When should you use in to and when should it be into?

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Comments  (Page 4) 
I came across some trouble while writing an article for class. If you could tell me which of the following is correct:

"Her ideas came in to play..." or "Her ideas came into play..."

Her ideas came into play.
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Dialing into the conference, can be considered figuratively entering the conference. One of into's many definitions is a literal or figurative entrance. Buying into an idea, is being involved with or interested in an idea. Involved with or interested in, is yet another definition for into. Turning into a bat, is a transition or state of change, which according to people on this site is another definition for into. The concept seems to be that it's almost always into as opposed to in to. The comments given imply it's only in to, when it means in order to. Additionally you'll notice a difference when sentences using into or in to are spoken aloud. The natural tendencies to speak the two words together is usually a good indication of when to use into. If you find there's a slight pause between the words, it's a pretty good indication that it's supposed to be two words.
Hey, actually He's Just Not That Into You is correct. Merriam-webster gives a plethora of definitions for INTO. One of those definitions is involved with or interested in. So "He's Just Not that Interested in You.", clearly falls under that definition, making the original title correct.
The answer is number 4. He ran into the bar.
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Would it be

The IT technician remotely dialed into/in to the computer.

The magician turned the hat into/in to a rabbit.