I'm really interested to know what his process was, since he didn't seem to actually know the product he was subtracting 50 from.

 Well it seems that he was always going for 4 from the top row, which reliably gives him 25, 50, 75 and 100 (I think). He then uses the ability to divide by 25 to treat these as an additional 2, 3 and 4. So once he has 318, he does (318 * 3) - 2 to get his 952, except via the 25s; ((318 * 75) - 50)/25. It's a hack; increasing the set of available numbers by guaranteeing the presence of a common divisor.
 And that's Numberwang!
 Well, obviously.
 I'd think of the thought process a bit differently than lclarkmichalek, though of course with the same result.To get close to 952, you can quickly think of 106x9. 106 is easy to obtain and you have a 3. You can get another 3 from 75/25. You're now at 954 with only a 50 left. If you could divide by 25, that would give you the 2 you're missing but you already used the 25, unless you were to divide later. So instead of doing 106x3x(75/25), you do (106x3x75-50)/25.He could have certainly thought of it another way but based on how players typically play that game, that would be a somewhat logical progression.
 Another way to think about it:He had the numbers 100, 3, 6, 25, 50, 75. 25, 50 and 75 are big and difficult to work with, but 50/25=2 and 75/25=3 are far easier. He could either do it right away, but that gives him either (2 and 75) or (3 and 25) and there's still a large number. (75x ± 50y)/25 on the other hand equals (3x ± 2y) and he's down to nice small numbers.
 I was wondering how he thought of that. Even though we don't know, this explanation is beautiful and makes a lot of sense. Thanks!
 There's two answers already that seem correct, but perhaps overcomplex, so here's my go.He always chooses 4 from the top row, so he always gets 25, 50, 75, 100 and the rest are chosen randomly.Using them in combination he can always trade 25/50 for a "2", 75/25 for a "3" and 100/25 for a "4" if he needs them to get the answer. Rather than work that out on the fly he just remembers it.Taking it once step further he can do (75x ± 100)/25 and get 3x ± 4, or (75x ± 50) / 25 and get 3x ± 2 if that would be helpful.One of the other answers points out that he can go further and multiply that 50 or 100 by any of the random numbers he's given, which would be equivalent to multiplying the ± constant by the same amount though he doesn't use that level of complexity in his answer.So he's basically building a toolbox of potential moves based on knowing that he'll always get those 4 numbers. He doesn't need to do the full calculation each time.

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