# What is the Sequence that Maximizes the Distance between my Wearing Two Similar Sweaters

Let’s say that I have 5 sweaters. A white one (Wh), a brown one (Br), a blue one (Bl) and two purple ones (P1 and P2).

Unfortunately, I’m also a very fussy person and I refuse to wear the same colour twice in a row. In addition, I want the sequence of sweaters to remain the same forever once I have decided which one is the best. However, I can wear the same sweater for 2-3 days before changing. In other words, I can go almost 3 weeks before wearing the first one again.

Considering that I only intend to wear them at work (Monday-Friday) what is the pattern that maximizes the distance between my wearing the two purple sweaters ?

I am no mathematician so I assume I must be missing something here.

If you wear them in any sequence but keep it the same once you get started, surely that will resolve the “problem”?

These two statements seem at odds. If you can wear the same sweater for 2-3 days, isn’t that the same as wearing it 2-3 times in a row?

I’m also not sure how these two statements are consistent. If each sweater can be worn for three days and there are five sweaters, then you can go 15 days (3x5) or just over two weeks before wearing the first sweater again - not “almost three weeks.”

I’d have to understand the question better and resolve the above seeming inconsistencies before I could try to answer it.

Get rid of one of the purple sweaters. Buy a green one.
Now you have a 5-sweater rotation.
Rotate.

mmm

Assuming the OP meant maximizing the distance between the two multi-day wearings of the purple ones …

Mathematically, you’re looking at a circular queue with 5 elements. The purple sweaters occupy 2 slots of the 5 and so you only have 3 remaining slots to separate them with. Once you decide not to put the two purple sweaters in adjacent slots, that means there must be one intervening sweater on one side and two intervening sweaters on the other. So purple-1, X, purple-2, Y, Z, repeat … Or equivalently purple-1, X, Y, purple-2, Z, repeat …

Once you start into your fixed rotation it doesn’t matter where you start.; the spacing will remain as I’ve described and that’s as good as it gets.

Separately …
Given the 2-3 days consecutive use rule and the days of the week, there is no way to ensure the rotation stays fixed versus the weekdays and weekends. So if you were considering that “Monday is farther from Friday than Tuesday is from Monday”, and trying to use that fact in your rotation, it won’t work.

Conversely, if you always wore each sweater only one day and you believed the above rule, you would in effect have 2 more non-purple days in a 7-day queue. Which means you could separate the two purple sweaters by 5 non-purple days arranged as one group of 2 and one group of 3.

That’s indeed what I meant.

I guess I worded this poorly, which caused the confusion above. In other words, I do not mind wearing the same sweater for 2-3 days. However, when I change, I do mind immediately wearing the same colour.

If I understand you correctly, that’s the same conclusion as the one I arrived at intuitively :

either
P1 - X - X - P2 - X
or
P1 - X - P2 - X - X

I sort of hoped there would be a more optimal mathematical solution that I wasn’t aware of. Still, thanks for your reply.

But I like both my purple sweaters !

How would that be possible with the sweaters you start with? If you can go 15 work days before wearing the same sweater, best possible solution would be 7.5 days between purples.

P1, Wh, Br, Bl

Repeat indefinitely. Never wear P2.

Or, consider this:

P1, Wh, Br, Bl, P2 (repeated) will simultaneously maximize (3) and minimize (1) the distance between P1 and P2 for this set of sweaters.

Better, since you specified the distance between purple sweaters only:

n(P1, Wh, Br, BL), P2 is the pattern as a whole, with n chosen to be however large you like while still allowing you to wear P2 every few years.