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Can anyone explain to me why this rule is set up this way?

*Grace helps move the sofa if, but only if, Heather helps move the recliner

if Gs --> Hr
if Hs/t -->Gr/t
if Hr -->Gs
if Gr/t --> Hs/t

Originally, I only wrote the last two lines in my chart and can not figure out why all four have to be written (Gs->Hr sounds wrong to me)... can anyone please-please explain that to me??

Question: Four people- Grace, Heather, Josh, and Marie will help each other move exactly three pieces of furniture- a recliner, a sofa, and a table. Each piece of furniture will be moved by exactly two of the people, and each person will help move at least one of the pieces of furniture, subject to the following constraints:.....

Thanks!

This is a biconditional, meaning that there are two possibilities: both conditions are triggered or neither one is triggered. To see why, break it into two pieces:
Grace helps move the sofa if Heather helps move the recliner. (HR ---> GS)
Grace helps move the sofa only if Heather helps move the recliner. (GS ---> HR)

Combined, this rule can be diagrammed as follows:
GS <---> HR

Be careful with those contrapositives. In this particular game, some of the people will be assigned to more than one piece of furniture, so you should instead reference the furniture pieces in the original rule like so:
~GS <---> ~HR

For instance, take a look at question 9. Here, we must assign an H to each of the three pieces. This triggers the conditional, and it must be true that G is assigned to the sofa. According to the contrapositive you have written, she would have to be assigned to the recliner or the table. However, since each of the four people must be assigned to at least one piece, and H is assigned to all three, G can only be assigned to one piece (the sofa) under this scenario.

azizi wrote:if Hs/t -->Gr/t

It's two rules:

If Grace helps move the sofa then Heather helps move the recliner.
Gs --> Hr
~Hr --> ~Gs

If Heather helps move the recliner then Grace helps move the sofa
Hr --> Gs
~Gs --> ~Hr

All responses thus far are true and correct. (Yes, 2 separate conditionals, one reflecting the "if" and one reflecting the "only if".) In my experience, people have relatively little trouble understanding how to diagram "if" clues, but significant trouble diagramming "only if" clues. Here's my solution.

Diagramming "Weird" conditionals:
Step 1: Identify the "weird" conditional by noticing the "weird" word. There are only two "weird" words, "Only" and "Unless".
Step 2: Look at the action that comes directly after the "weird" word in the clue.
Step 3: Begin your diagram by negating that action that you just noticed.
Step 4: Brain it out.

Example: Jake won't go the party unless Sarah goes to the party. Step 1: ID: Unless! Step 2: Notice: Sarah goes to the party. Step 3: Negate. Sarah does not go the party. Step 4: Brain: What do I know to be true if Sarah does not go to the party? Well, Jake won't go. Diagram: ~S --> ~J. Contrapositive: J --> S

*Don't forget. In Step 3, when you negate the action that comes after the weird word, remember that when you negate an "and" it becomes an "or" and vice-versa.

I know this is relatively simple and well explained elsewhere. But I've just always been more comfortable thinking things through and completely understanding what I'm doing rather than simply memorizing that "only" and "unless" make the conditional go backwards or some such LSAT robot advice.