# The half-full cup of germs

You are at a large international microbiology conference, having small talk with fellow researchers, when someone raises the famous old riddle:

"If, starting from one bacterium it will take a 100 time units to have a cup-full of bacteria, how long would it take to get a cup-full if we start with two bacteria?"

"50 time units!" you immediately say, trying to blend in the conversation (did I mention you are EXTREMELY shy, have a very hard time making small talk, and don't know anyone here?). You immediately realise the mistake you've made, everyone knows that the correct answer is 99 time units since bacteria usually replicate themselves every time unit.

You start getting weird looks, you think you hear someone murmur "how stupid" (or maybe it was "god bless you"?) someone says, "did you ever see an example for this?" and for some reason you say "yes!", and now there are follow-ups. "Elaborate?"

Quick! how do you explain why one bacteria creates a cup-full in 100 time units and two create it in 50?

Keep in mind the following:

• The number of new bacteria every time unit depends ONLY on the number of them in the previous unit.
• There's no such thing as a part-bacterium.
• The number of bacteria in a full cup is orders of magnitude larger than 1.
• Hi Dotan, I just changed your 'microbiology' tag to the more general 'science' since we usually only create new tags for Puzzling once we have a significant number of puzzles that could use them... :) – Stiv Apr 12 at 10:25
• Is this a [lateral-thinking] puzzle, or is this actually about inventing a plausible method by which a strain of bacteria could produce these results? – Magma Apr 12 at 15:50
• "half full cup of germs" ... is this one case where saying half full is actually the pessimistic option? – Rand al'Thor Apr 13 at 5:05

Directly ripping off @Pere's answer, just rewording for OP's technicality. Pere if you'd like to steal any part and add to yours - please do! Debated a comment instead.

This strain of bacteria is typically unable to reproduce until it has undergone bacterial conjugation. It's very close to sexual reproduction in that sense, as the conjugation triggers the reproductive process on completion. The bacteria evolved this in order to keep mutations controlled, and even help positive mutations spread in the population.

However, if a bacterium can't find any other bacterium to reproduce with for 50 time units, it then undergoes division without bacterial conjugation. After that, the resulting bacteria start reproducing exponentially until they get the cup full in 50 time units.

Starting with 2 units, we don't need to wait for exponential growth, it starts 50 time units earlier, and are finished in 50 total time units.

• Technically this solution does in fact violate the rule that "The number of new bacteria every time unit depends ONLY on the number of them in the previous unit." - in the first scenario, we have the same number of bacteria (1) at the moments $t=0$ and $t=49$, but the result at $t=1$ and $t=50$ differs. In fact, the number of new bacteria depends not only on the number of bacteria at current time but also on their state. – trolley813 Apr 17 at 16:44
• But their state is determined by the number of bacteria – TCooper Apr 17 at 19:06
• So, is there no difference between a single bacterium at the very beginning and the very same bacterium after 49 time units? – trolley813 Apr 17 at 20:07
• Of course, I agree that the problem is not in the answer but in the question. – trolley813 Apr 17 at 20:08

This strain of bacteria always reproduce by bacterial conjugation, for which at least two bacteria are needed. However, if a bacterium can't find any other bacterium to reproduce with, it undergoes division. A bacterium needs 50 time units to "realize" that it's alone and divide. After that, the resulting bacteria start reproducing exponentially until they get the cup full in 50 time units.

If we start the experiment with two bacteria, exponential growth starts 50 time units earlier.

• Very nice! but I can't accept this since conjugation is a form of horizontal gene transfer that doesn't include reproduction... find different wording maybe – Dotan Apr 13 at 6:05
• Yes, I must admit that I stretched a lot what sexual reproduction in bacteria is and my answer could have fit better a question about some kind of extraterrestrial aquatic plan than one about bacteria. – Pere Apr 13 at 9:36

The bacterial population is described by a logistic curve, which has the property of reaching half its eventual maximum halfway through the growth, and is also a much more accurate model for bacterial growth in a Petri dish.
Placing the two bacteria in different cups will give two half-cups after 50 time units, so 1 cupful total. Image credit: "Environmental limits to population growth: Figure 1" by OpenStax College, Biology, CC BY 4.0.

• This almost works: if you start with 2 bacteria you're very near the left side of the curve so it will still take you the same amount of time to reach the end, even though you'll be close to filling the cup after half the time (which was true even with one bacterium) – Dotan Apr 13 at 6:02
• Start with both bacterium in separate cups and combine them after 50 time units. – InternetHobo Apr 13 at 16:40
• @Dotan: Yes, what InternetHobo said is what I had in mind. – Nick Matteo Apr 13 at 19:36

They're in two different-sized cups.

• If the bacteria are indeed doubling each time unit, this would only matter if the cup's volume was less than a trillionth the size of the original cup. – Nick Matteo Apr 13 at 2:44

Well, rather artificially and implausibly we could have the following setup which meets all the explicit requirements in the puzzle:

If there are N bacteria in a given place at time t, then at time t+1 there are 3N if N is odd and 9N if N is even. So if you start with an odd number you always have an odd number and they reproduce "slowly", and if you start with an even number you always have an even number and they reproduce "quickly".

Of course this

is physically preposterous. A colony of bacteria couldn't possibly "know" whether it was odd or even in size, and the number of bacteria is absurd anyway. And the growth rates are ... implausible. A "fast" colony starting with two bacteria, even if each has only the mass of a proton, will be approaching the mass of the earth after 50 time units.

• Actually, in your case, if 1 bacterium can fill a cup in 100 units, then 2 bacteria can fill two cups in 50 units. – trolley813 Apr 12 at 11:13
• Indeed! But the time to fill one cup is also 50 units. – Gareth McCaughan Apr 12 at 11:40

The strain of bacterium used for this particular experiment replicate every time unit, but the propagator then becomes sexually inactive, and breeds no more. Therefore the growth rate of said bacterium is linear, and two bacterium are better than one!

Or:

You realise you have made a mistake, and explain that you read $$2^{100}\cdot\frac12$$ as $${2^{100}}^{\frac12}$$.

• Small cup! (Or big bacteria!) – Stiv Apr 12 at 11:28
• @Stiv; I think by 'cup' OP means en.wikipedia.org/wiki/Petri_dish, and so not so impossible. – JMP Apr 12 at 11:30
• I think this is inconsistent with the last stated condition: "The number of bacteria in a full cup is orders of magnitude larger than 1.". – Gareth McCaughan Apr 12 at 11:41
• @GarethMcCaughan; the propagator doesn't die, so there are 100 bactum's at $t_{100}$. – JMP Apr 12 at 11:47
• While strictly speaking that's "orders of magnitude larger" I don't think it's at all what OP intended the condition to mean. – Gareth McCaughan Apr 12 at 13:53

"If, starting from one bacterium it will take a 100 time units to have a cup-full of bacteria, how long would it take to get a cup-full if we start with two bacteria?"

Nowhere does it say the bacteria doubles in a time unit.

We could have the bacteria double every $$50$$ times units. (Or alternatively, if the bacteria do double in a time unit, we just use a time unit that is fifty times shorter.)

In which case a cup with $$1$$ bacterium will fill up to a cup full of $$4$$ bacteria in $$100$$ time units and a cup with $$2$$ bacteria will fill up to a cup full of $$4$$ bacteria in $$50$$ time units.

BUt that is a very small cup.

The formula for exponential growth would be the number of bacteria after $$t$$ time units is $$N(t) = 1*b^t$$ for some base $$b$$. Nowhere is it required that $$b = 2$$.

We must have $$N(100) = b^{100}=C$$ a full cup full. And we not that $$b^{50} = \frac 12 C$$. Hence we have $$b^{50}= \frac {b^{100}}{b^{50}} = \frac {C}{\frac 12C}=2$$ and $$b = \sqrt{2}$$.

So To have $$N(100) = C$$ and $$N(50) = \frac 12 C$$ we must have $$b=\sqrt2$$ and $$C = 4$$.

• Nice guess, but still, the problem body states that 'The number of bacteria in a full cup is orders of magnitude larger than 1.' – newbie Apr 12 at 23:57

Even more artificial case:

Let $$b_n$$ be the number of bacteria after $$n$$ time units. Let's define a recurrent relation: $$b_{n+1}=(b_n+1)\mod99+1$$ if $$b_n$$ was odd and $$2b_n$$ otherwise (i.e. linear growth when the current number of bacteria was odd and exponential when even). So, starting with $$b_0=1$$, we get $$b_1=3$$, $$b_2=5$$ etc. up to $$b_{49}=99$$, then $$b_{50}=2$$ (after 50 time units starting with 1 bacterium, we get 2 bacteria — this means that 2 bacteria can fill the cup in 50 units if 1 can do it in 100), then $$b_{51}=4$$, then 8, 16 etc. up to $$2^{50}$$. Well, this is even more unrealistic than Gareth McCaughan's answer, but the resulting number is somewhat real (about $$10^{15}$$).

• The bacteria is too clever :D – newbie Apr 12 at 12:05

Not a mathematics answer, but a science one:

Bacteria have limited lifespan. They start to die at some point (and may even have different living time).

• For most things that have both birthrates and deathrates, this doesn't matter; the overall growth rate is still exponential. For the deathrate to matter, you'll still need to find some way for this bacteria to be special. – Brilliand Apr 14 at 19:13
• @Brilliand I'm just pointing out one possibility, which hasn't been mentioned. Why do you assume that the death rate is constant? Have a look at the other answers - are they more realistic or scientifically correct than mine? – WhatsUp Apr 15 at 1:01
• @WhatsUp That's a good point, now let's see if you can use it to answer the question? – Dotan Apr 15 at 7:14
• @WhatsUp You'll still need to find some way for this bacteria to be special. Pointing out that there probably is a way isn't good enough; you have to find what that way is. – Brilliand Apr 24 at 22:35