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## A Valentine’s Day Logic Puzzle – WALKTHROUGH

**STEP ONE:**

From the start, we see that Alan & Beth are celebrating Valentine’s Day together. This means that nobody else is celebrating with Alan, and nobody else is celebrating with Beth (since each individual is in only one relationship). Thus, we put X in the rest of Alan’s and Beth’s columns.

Whenever we successfully match a couple, we will need to do this for all remaining boxes in the matched row and column.

**STEP TWO:**

We also know that Sue & Steve and Mike & Megan cannot be together, since there is only one alliterative couple (which is Ryan’s). We put an X in those boxes. Also from the first clue, we see that Ryan is the only person seeing someone whose name begins with the same first letter. So Ryan is not dating Kate, Mike, or Sue (since they don’t begin with R).

Since Sue is heterosexual, she must be dating a guy. Thus she is not dating either Erica or Megan; Xs go in those boxes.

**STEP THREE:**

The only person left who Sue can be with is Leon. We put R in that box.

Since Leon is spending Valentine’s Day with Sue, no one else can be with Leon. Fill in Xs in the Leon’s remaining boxes.

**STEP FOUR:**

The end of this new clue states that Rick is in a same-sex relationship. Therefore Rick is not seeing either Erica or Megan (put Xs in those boxes).

**STEP FIVE:**

This is the toughest part of the puzzle.

Consider there being only one same-sex relationship:

– This would mean that Rick is in a relationship with another man.

– There would thus be 2 women and 1 man remaining in both the rows and the columns. With these options left, it is not possible to create 3 heterosexual couples (because the number of women in the rows, does not equal the number of men in the columns).

– Therefore, there MUST be 2 same-sex relationships.

– Using the same reasoning as above, the second same-sex relationship must be between 2 females (since Rick is in an all-male relationship).

Now that we know there must be a same-sex female couple, we re-examine the grid. Megan has only females remaining that she can be dating (Kate and Rachel). Therefore, Megan must be in the other same-sex relationship.

Thus, we know that Erica is NOT spending Valentine’s Day with a woman. This means Erica must be seeing Mike. We then put Xs in Mike’s and Erica’s remaining open boxes.

**STEP SIX:**

Steve is celebrating with a woman, so he is not seeing Rick. This means the only person left who Rick can spend Valentine’s with is Ryan. We put an X in Ryan’s remaining open box.

**STEP SEVEN:**

Only one of Kate and Rachel have a name that comes alphabetically before their partner’s.

– “Rachel” comes after “Megan”, but before “Steve”.

– “Kate” comes before both “Megan” and “Steve”.

Therefore, Kate must be the one who satisfies the condition in the newest clue. Consequently, Rachel must be dating Megan (because “Rachel” comes after only “Megan”), which means Kate is spending Valentine’s Day with Steve.

**STEP EIGHT:**

Happy Valentine’s Day!