Puzzle

Challenge 1

Sarina needs 14 different outfits, since she plans to change outfits once each day. So she needs at least two tops and two bottoms. Let’s call these T1, T2, and B1, B2.

Construct a table listing all the possible outfits she can wear with two tops and two bottoms.

Two tops and two bottoms provide only four possible outfits.

Let’s experiment by keeping the number of tops constant and see how many bottoms Sarina needs.

With two tops and three bottoms we get six outfits:

Following through with this pattern, we see that Sarina can have 14 different outfits if she takes seven bottoms to go with her two tops. So she has to pack nine garments in all.

Sarina wants to pack the smallest number of total garments. Is there any way she can have 14 different outfits with a smaller total number of garments?

By this time you’ve probably figured out that the number of outfits is the product of the number of tops and bottoms. So if Sarina packs either three tops and five bottoms or five tops and three bottoms, she’ll have 15 different possible outfits. That’s a total of eight garments she has to pack.

Three tops and five bottoms

There’s one more way Sarina can have enough outfits with eight garments: four tops and four bottoms make a total of 16 different outfits.

So we have found three different ways that Sarina can pack eight garments so that she can vary her outfits twice a day and never wear the same outfit more than once in a week.

Challenge 2

For this challenge Sarina plans to change from pants to skirt, or vice versa, every day. The reasoning is similar, but with a different constraint, the results may change. For example, Sarina needs seven different skirt-blouse outfits and seven different pants-blouse outfits.

As before, Sarina needs at least two tops so she can change tops once a day. To get seven outfits, she can pack four skirts and two tops for eight skirt-blouse outfits and four additional pants to make eight pants-blouse outfits. Four skirts, four pairs of pants, and two tops make ten garments all together.

Can she do it with fewer garments overall?

Suppose Sarina packs three tops. Then she’ll need three skirts (nine skirt-blouse outfits) and three pairs of pants (nine pants-blouse outfits). This is a total of nine garments, in the exercise above. This is one less than with two tops and four of each kind of bottom. But can she do it with fewer than nine garments?

Suppose Sarina packs four tops. Then she needs only two skirts and two pairs of pants to have eight of each type of outfit. Four tops plus two skirts and two pants equal eight garments in all.

Therefore the smallest number of total garments is still eight! But we found three different ways to have eight garments for Challenge 1. There is only one way to do it for Challenge 2.