The SAT will throw two common kinds of “counting” problems your way. I’ll handle one of them in this post. The other kind, well, I’ll get to it when I get to it. 🙂

I like to call this kind of problem a matching problem. It’ll usually involve a bunch of people who all need to shake hands, or a league in which every team needs to play every other team. Or a massage club where everyone has to give everyone else a back massage. I don’t know…whatever. Everyone has to touch everyone at the touching party?

…this is going nowhere. Let’s see an example.

  1. Each team in a kickball league plays each other team 4 times during the season. If there are 7 teams in the league, how many games long is the season?
    (A) 28
    (B) 56
    (C) 84
    (D) 112
    (E) 116

What you’re going to want to do here is draw a diagram.

Arrange the letters A-G (representing the 7 teams) in a large circle. Now draw
lines connecting each letter to each other letter, carefully counting as you draw. (If you try to count after you’re done drawing, you’re going to have a pretty difficult time getting an accurate count.) The best way to go about this is to draw every line that you can that originates at
A, and then do the same for B, etc.

You’ll know you’re done when you have something that resembles a star with all its outer points connected. Like so:

The number of lines you just drew – 21, you awesome counter you – equals the number of games required for each team to play each other once. If each team has to play each other 4 times, multiply 21 by 4 to get the answer: 84! BAM!

Pretty amazing, right? It’s just so…beautiful. No, stop crying. It’s totally inappropriate for you to be crying right now. I know it’s pretty but you need to stop. I refuse to move on until you stop crying.

Of course, if you want to represent the above diagram mathematically, you could say that Team A needs to play all the other teams, so it plays 6 games. Then Team B needs to play all the teams except Team A (since they already played), so that’s 5 more games. Follow that line of reasoning until its end and you get:

6 + 5 + 4 + 3 + 2 + 1 = 21

But I think the star is prettier (and easier to remember).

Care to try a few more?

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