Maximum number of mathematicians in this joke

[Johnny_L.A]

@Prof.Pepperwinkle

An infinite number of mathematicians walks into a bar. The first one orders a beer. The second one orders a half a beer. The third one orders a quarter of a beer. The fourth one orders an eight of a beer…
The bartender pours two beers, and says, “You figure it out!”

Funny joke, but an infinite number of mathematicians is too many. There are only so many molecules in a pint of beer. To make it even more complex, there aren’t ‘beer molecules’, but a number of molecules that, in combination, constitute ‘beer’.

Given one pint (473.176 ml) of a given beer, between how many mathematicians may it be divided before 1) a hypothetical 'beer molecule cannot be divided; or 2) any more divisions would create a substance that does not constitute ‘beer’ because one of the ingredient molecules is missing?

[pjd]

What brand of beer are we talking about here ?

[Johnny_L.A]

I was going to say ‘Let’s call it Guinness, because I like Guinness’ in the OP, but didn’t.

[F.U.Shakespeare]

Maybe there should also be some chemists and physicists.

[TriPolar]

Why did the bartender pour 2 beers?

[F.U.Shakespeare]

One on the house?

[pjd]

136 would probably be enough.
…or 142 maybe.

[markn_1]

Beer is over 90% water, so to a first approximation we can just assume it’s all water. 473 ml of water weighs 473 grams, and the molecular weight of H₂O is 18 g/mol, so the pint contains 473/18 = 26.3 moles = 1.6 x 10²⁵ molecules. That’s about 2⁸⁴, so the 85th mathematician gets one molecule and everyone after him has to go to another bar.

[Prof.Pepperwinkle]

@Johnny_L.A

Funny joke, but an infinite number of mathematicians is too many. There are only so many molecules in a pint of beer. To make it even more complex, there aren’t ‘beer molecules’, but a number of molecules that, in combination, constitute ‘beer’.

Given one pint (473.176 ml) of a given beer, between how many mathematicians may it be divided before 1) a hypothetical 'beer molecule cannot be divided; or 2) any more divisions would create a substance that does not constitute ‘beer’ because one of the ingredient molecules is missing

Ah, but an infinite number of mathematicians, skilled in the use of irrational, unreal, transfinite and imaginary numbers, can always find a way to throttle an answer out of such a situation.

[Dewey_Finn]

@TriPolar

Why did the bartender pour 2 beers?

Because the first person received a full glass, the second received half, the third a quarter and so forth. It should add up to two glasses.

[F.U.Shakespeare]

I wish to apologize for cracking wise before the question had been answered. One doesn’t often get the opportunity to make math jokes.

[dtilque]

@TriPolar

Why did the bartender pour 2 beers?

Explaining jokes always ruins them, so here’s the ruined joke:

1 + 1/2 + 1/4 + 1/8 + … = 2

[Ninja’d but I’ll post anyway.]

[TriPolar]

Doh! I assumed the bartender had already poured the first four beers.

[Deeg]

I wanna know how all those mathematicians fit into the bar in the first place.

[markn_1]

It was the bar in Hilbert’s Hotel.

[Chronos]

@Johnny_L.A

or 2) any more divisions would create a substance that does not constitute ‘beer’ because one of the ingredient molecules is missing?

Others have handled the first part, so I’ll tackle this one. To a second approximation, beer consists of water and ethanol. Guinness is 4.2% alcohol by volume, which translates to 3.4% alcohol by weight. Ethanol has a molecular mass of 46, compared to water’s 10, so it has about 130 water molecules per ethanol molecule. That’s very nearly 2^7, so requiring each mathematician to have at least one ethanol molecule reduces the number of possible mathematicians by 7.

[markn_1]

@markn_1

473 ml of water weighs 473 grams, and the molecular weight of H2O is 18 g/mol, so the pint contains 473/18 = 26.3 moles = 1.6 x 10²⁵ molecules. That’s about 2⁸⁴, so the 85th mathematician gets one molecule and everyone after him has to go to another bar.

Correction: as Chronos says, water’s mass is 10, not 18, so there are 47.3 moles = 2.8 x 10²⁵ = ~ 2⁸⁵ molecules , which actually doesn’t materially change the result.

(Just noticed that Discourse removes the < sup > tags when you quote text, so I manaully added them back. That sucks.)

[Chronos]

Correction to correction: Water’s mass is, in fact, 18. Oxygen has 8 protons, but then also that many neutrons. I had it wrong before.

[TroutMan]

@Chronos

To a second approximation, beer consists of water and ethanol. Guinness is 4.2% alcohol by volume, which translates to 3.4% alcohol by weight.

We need a better definition of “beer” than water and ethanol, because water with 4.2% v/v ethanol is just watered-down vodka. In order to be considered beer, there needs to be at least some “malt” molecules and “hop oil” molecules, in the proper balance to create the taste of beer.

We can’t ignore those just because they’re in small amounts; in fact, the smallest contributor that must be present to create “beer” will determine how many parts the pint can be split into.

[Chronos]

And actually, it gets much worse: Most of the other compounds in beer are going to be much larger molecules than water or ethanol. Which means that the requisite single molecule of one of those will be equal to many water and ethanol molecules, and so will need many more to make the proper proportions.