Nervous Bathroom Break

To get the minimum number of people using the urinals without being next to each other, we need to get the maximum distance between two persons without giving room for another person between them. For example, if a person is four urinals away from another, a third person can still use the urinal between them without being next to either person.

Hence, the ideal distance is 3 urinals apart. So if we have the first person on one end of the row, that will leave us with 99 urinals to divide by 3. That is 99 / 3 + 1 = 34

I used Photoshop to solve this visually. I created a canvas of 100 by 1 pixels, hence simulating the 100 urinals (one pixel = one urinal). Then, using green pixels as an occupied urinal and red pixels as all adjacent urinals, I filled out the row as such:

After that, you can either count the green pixels, or cheat and use a layer counting script to get the answer (this relies on you creating each layer being a red-green-red 3 pixel set to represent a person). By doing so, you obtain the answer 34.

As a generalization, it can be said that the minimum amount is $\lceil{\frac{x}{3}}\rceil$ where $x$ is the amount of urinals, since every third urinal can be occupied thus making anyone who goes between those urinals next to someone.

Well starting with the first urinal the next person stands at fourth urinal, then 7th urinal, then tenth and so on keep adding 3 urinals, till you reach hundred. So that the gap between two persons is ONLY two urinals, By, this combination two things happen:

1) Maximum urinals are covered with minimum persons.

2) It satisfies the condition for the 35th person, to forcefully urinate next to other person.

NOTE: Two person standing with only one urinal difference is the best combination here, and takes 50 persons, before anybody is forced to pee besides someone else.

Really simple, just test it. A hundred is a big number, let's try to apply with smaller numbers.

A single person can invalidate up to 3 urinals. (chosen one plus sides)

Up to 3 urinals, a single person is enough to force someone to come to their side. At 4 urinals, the bathroom needs 2 guys

5 needs 2, 6 needs 2, 7 needs 3, 8 needs 3, 9 needs 3, 10 needs 4...

You can keep on, but the pattern is already obvious, each 3 urinal we add, one more person is necessary, this can be translated to an equation, this way you can test in a more specific way to find out: f(u) = u/3 (The ONE is because we need one at the beginning)

And we can now test before putting the number of urinals we want

f(1) = 1/3 = 0.33 => round UP to get 1

f(3) = 3/3 = 1.00 => round UP to get 1

f(4) = 4/3 = 1.33 => round UP to get 2

f(7) = 7/3 = 2.33 => round UP to get 3

Alright, we went through extra work just to make sure our answer would be correct!

f(100) = 100/3 = 33.33 => round UP = 34

If every urinal save one was filled, you would need 99. If every urinal was filled save for two, the next person would need to stand next someone still, and you'd need 98. In the case of 97, if all three open urinals were next to each other, this would leave a "non adjacent" free urinal. So, the max free urinal chain can be two. To cover the ends of the urinal line, as the urinals there can only be "blocked" on side, you need one urinal filled right next to each end. That takes up 4 urinals, leaving 96. From there, using the two rule, adding a person to "block" adjacent urinals, means that chains of three are there on filled. So, 100 - 4 =96 and 96/3 = 32. Add the end filled urinals and the answer is 34