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Today we’ll be working a game problem from the June 2007 LSAT. You’ll need to download a copy from the LSAC website and you should probably print it out. We’ll be starting with game 1 on page 4.

The first step in any game is to read the game and figure out what it is we’ll actually be asked to do. In this case we’re asked to put numbers in order to build a product code. The tricky part is the code will have five digits, but we’ll be using the numbers 0-4, so make sure you keep that in mind. 

Next we’ll need to figure out how we want to represent the game. Since we’re putting numbers in order, the best thing to do will be to make a diagram for with places 1st, 2nd, 3rd, 4th, and 5th being the columns. I’m using the ordinal numbers 1st, 2nd, etc to avoid confusion with our inventory of 1, 2, etc. Remember, our diagram is always going to be the thing we’re placing the inventory into, and in this case we’re placing the digits 0-4 into order 1st-5th. It’s always best to make your diagram under the question column with the most room, so in this case I’ve made my diagram under the second column of questions and reserved the space under the first column of questions for my rules and deductions. We’ll put our inventory of digits 0-4 above the diagram a bit to remind us what we need to place.

Diagram and more after the jump.

Now we’re ready for our rules:

Rule 1. There doesn’t seem to be much to this rule and so it probably isn’t very easy to symbolize, so let’s boil it down. It’s telling us that we’re only going to use 0-4 and no other numbers, so I’ve shortened that to “0-4 only”. Remember even if it’s a simple rule like this, we want to write all of them out.

Rule 2. This one is similarly simple but gives us a very important piece of information: we’ll be using each number once and since we’ll be using each only once, that also means that we’ll have to use all of them as well. So I’ve written “0-4 only once each”

Rule 3. I’m not a fan of using algebraic notation (or anything looking like math) in a game, but in this case we’re given a rule that’s pretty explicitly mathematic, so we’ll have to compromise. The rule tells us that the 2nd digit must be exactly twice the value of the 1st. So I’ve got “2nd = 2 x 1st”.

Rule 4. Same issue as rule 3, I’d rather avoid mathematic notation, but again, we’re dealing with numbers here, so I’ve got “3rd < 5th”. If you get confused about less than vs. greater than just think of the sign like a mouth, the open part always wants to eat the bigger number, so this reads, “5th is greater than 3rd”

Rules are done.

Now, most importantly, double-check. Make sure you’ve got each rule written correctly and you understand how you’ve symbolized or shortened it.

Now we get to the deductions. Our deductions here revolve around rules 3 and 4. With a rule like rule 3 you want to make sure you always try to plug-in possible values to see if you can narrow things down a bit. If the 2nd digit has to be double the 1st we should try plugging in some possible values for the first and see what happens. If we start with 0 we’ll see that won’t work (double 0 is still 0), so that’s out. 1 doubles to 2, so that’s a possibility. 2 doubles to 4 and works. 3 doubles to 6, which is outside the values of our inventory, and we’ll have the same problem with 4. So we know that the only possible values for the 1st digit are 1 and 2, and the only possible values for the 2nd digit are 2 and 4, respectively. That certainly helps restrict things.

Likewise, rule 4 should allow us to restrict the game a bit further. Anytime something has to be less than or greater than another thing it immediately rules out some possibilities. We know that the 3rd must be less than the 5th. So, could the 3rd digit be 4? Well, if it was than the 5th digit would have to be greater than 4, which won’t work since 4 is the highest we go. Now we’ve ruled out 4 as a value for the 3rd digit. Similarly, could the 5th digit be 0? If we plug it in, than the 3rd would have to be less than zero, which while a pretty cool movie, isn’t one of the numbers in our inventory, so that’s out as well.

You’ll notice in my work shown below how I’ve laid everything out. I’ve got my deductions underneath my rules, jotted down, and I’ve got a #0. line in my diagram where I’ve put the things I know must be from my rules and deductions.With all that done, we’re ready to tackle the question.

Let’s start by drawing a new line for #1 in our diagram.

Question 1 asks us what must happen if the 5th digit is 1. First step is to put 1 under our 5th spot in our diagram. Now let’s start going through the rules. Rule 2 tells us we won’t be using 1 again. So we want to look first for any rules that deal with our inventory item (1 in this case), which there aren’t any, and next for rules that deal with our spot (5th). Bingo! Rule 4 tells us that 3rd<5th, so the only digit less than 1 is 0, so the 3rd digit must be 0.

Next we can look at our #0 line in the diagram. We see that the 1st column can only be 1 or 2. Well, 1 is taken by the 5th column so that leaves us with 2 for the 1st spot. With 2 in the first spot we know the 2nd spot must be 4 (rule 3 and our #0. line). Now the only thing inventory item left is 3 which will go in the 4th column.

Diagram finished, time to answer the question. We always check all the answer choices to ensure we didn’t make a mistake in our work, but with a fully completed diagram we know we’re probably in good shape. The only answer that matches with our diagram here is A, which is the credited response.

See, that wasn’t so bad.

I’ll post some more game questions and sample work in the coming weeks. My first full test guide should be up soon and will include in-depth instructions on how to approach any game. In the meantime, if you have questions about the LSAT or need help, contact us or check out the  forums!