
Potato paradox - Panoramix
https://en.wikipedia.org/wiki/Potato_paradox
======
ot
This is not that uncommon when optimizing code.

Your program is slow so you profile it, and find out that f() takes 99% of the
time. So you work a lot to optimize f(), and re-profiling shows that now f()
takes 98% of the time.

Doesn't seem that impressive after all the work you've put into optimizing
f(), but your program is actually twice as fast :)

~~~
wobbleblob
Without knowing your code, I can think of a performance hack to get the
running time of f() down to less than 10% of the time.

~~~
JupiterMoon
What is this?

~~~
andrelaszlo
Slowing down the rest of the program? :)

~~~
wobbleblob
Well yeah, just add a wait loop somewhere else

~~~
eCa
I thought they were speed-up loops...

[http://thedailywtf.com/articles/The-Speedup-
Loop](http://thedailywtf.com/articles/The-Speedup-Loop)

~~~
ryangittins
Hah, this is great! It reminds me of the classic

    
    
      static char buffer[1024*1024*2];
    

story from a few years back.

[http://www.dodgycoder.net/2012/02/coding-tricks-of-game-
deve...](http://www.dodgycoder.net/2012/02/coding-tricks-of-game-
developers.html)

~~~
stcredzero
From CToGD:

 _In other words, when a project gets handed down from above to launch in,
say, 3 months, there 's no way in hell you can get the servers requisitioned,
approved, and installed in that time. It became standard practice for each
team to slightly over-request server capacity with each project and throwing
the excess hosts into a rainy day pool, immediately available and
repurposeable as required._

I know of a Social Security Administration acquisition back in the days of 100
MHz processors. The bureaucracy took so long with this, by the time the order
could be put out to suppliers, 100 MHz processors were no longer available, so
SSA ended up with a bunch of workstations that were 166 MHz processors down-
clocked to 100 MHz. (Otherwise, they would have had to start the whole process
over again.)

------
chestervonwinch
Neat. This bumps up my list of food-related maths from 3 to 4. So far:

[https://en.wikipedia.org/wiki/Ham_sandwich_theorem](https://en.wikipedia.org/wiki/Ham_sandwich_theorem)

[https://en.wikipedia.org/wiki/Pizza_theorem](https://en.wikipedia.org/wiki/Pizza_theorem)

[https://en.wikipedia.org/wiki/Layer_cake_representation](https://en.wikipedia.org/wiki/Layer_cake_representation)

~~~
gjm11
One of the people I went to university with had a little (very short) mental
catalogue of "chromatic mathematical fruit jokes". There are exactly two
famous ones. "What's purple and commutes?" "An abelian grape." And: "What's
yellow and equivalent to the axiom of choice?" "Zorn's lemon." He invented
another, which requires more esoteric knowledge: "What's green and determined
up to isomorphism by its first Chern class?" "A lime bundle." I don't remember
whether anyone found a credible fourth example.

(Oh, how we laughed.)

[In case anyone reading this thinks the above might be amusing if only they
knew what mathematical objects were actually being referred to: (1) No,
probably not. (2) Abelian group; Zorn's lemma; line bundle. For the last one,
you need to make it the first _Stiefel-Whitney_ class if you're working over
the real numbers rather than the complex numbers.]

~~~
martincmartin
I vaguely remember a joke about limit of the supremum (lim sup) involving lime
soup. In fact, I brought some lime soup into math class one day in undergrad.
It tasted awful. And I can't remember the joke.

~~~
frogpelt
You don't need to. This is hilarious.

------
eck
A much more important example of this than "martian potatoes" is uranium
enrichment.

Natural uranium is ~1% U235; bombs need 90+% U235. So when you've enriched it
from 1% to 2% it doesn't _seem_ like you've made a lot of progress towards 90.

If instead of enriching U235 you think of it as eliminating U238, though, then
you've done half of the work.

~~~
cperciva
_If instead of enriching U235 you think of it as eliminating U238, though,
then you 've done half of the work._

That's the wrong way to think of it though. The right way to measure progress
is in terms of Separative Work Units:
[https://en.wikipedia.org/wiki/Separative_work_units](https://en.wikipedia.org/wiki/Separative_work_units)

If you start with 10000 kg of natural (0.7%) uranium and you want to separate
it into 45 kg of highly-enriched (90% U235) uranium and 9955 kg of depleted
(0.3% U235) uranium, then you will have to do 8800 kg of "Separative work
units".

On the other hand, separating that same fuel into 3000 kg of partially-
enriched (1.4% U235) uranium and 7000 kg of partially-depleted (0.4% U235)
uranium only takes 1790 kg of "Separative work units", even though the
increased concentration of U235 means that "half the U238 has been
eliminated".

Isotope enrichment is an area where, to borrow a line from software
engineering, the first 90% takes 90% of the time, and the last 10% takes the
other 90% of the time.

~~~
mhartl
_45 kg … and 9960 kg_

You mean 40 kg and 9960 kg (so the sum is 10000 kg)? _< insert "Were you a
Putnam Fellow?" joke here>_

~~~
disillusioned
Annnnd just spent an hour reading that delightful Colin Percival Putnam
thread. Thanks for the reminder on that. Man, people really loved to hate on
Tarsnap because of his perceived arrogance.

~~~
blacksmythe

      >> people really loved to hate on Tarsnap
    

On the contrary, Colin is highly regarded on HN both for his technical
ability, and his success in business (despite breaking from conventional
wisdom in how to be successful in business).

------
kzhahou
Another angle on this problem: How much water must you add to the potatoes to
make them 100% water?

Of course, you can add all the water in the universe and they'll still not be
100% water. The water-percent increment just gets smaller and smaller, the
more water you add.

This "potato paradox" illustrates the same effect, but in the other direction,
where a small relative decrease yields a large absolute decrease.

~~~
miciah
If you add water to the potato infinitely (hypothetically speaking), does the
water percentage approach 100%? And also, does the solid percentage approach
0%?

~~~
nightcracker
No.

You won't have a potato anymore. But you won't have water anymore either.

You'll have a black hole.

~~~
ajuc
Is potato-blackhole distinguishable from same mass water black-hole?

~~~
cbd1984
> Is potato-blackhole distinguishable from same mass water black-hole?

[https://en.wikipedia.org/wiki/No-
hair_theorem](https://en.wikipedia.org/wiki/No-hair_theorem)

We don't know if black holes have hair, so the answer to your question is "We
don't know".

[https://en.wikipedia.org/wiki/Black_hole_information_paradox](https://en.wikipedia.org/wiki/Black_hole_information_paradox)

------
ThrustVectoring
The solution is much more intuitive if you use odds ratios instead of
percentage probabilities. You go from a 99:1 ratio to a 98:2 (or 49:1) ratio.

In other words, it's another way of phrasing that it takes twice as much
evidence to be 99% sure as it is to be 98% sure. Or that it's twice as hard to
have 99% uptime than 98%.

~~~
philh
> it takes twice as much evidence to be 99% sure as it is to be 98% sure.

Not twice as much evidence. Evidence needs to be measured logarithmically.
(Otherwise you'd say it takes twice as much evidence to be 67% sure as 50%
sure (2:1 versus 1:1), but the second takes no evidence at all for a binary
proposition.)

It takes twice as much evidence to be 99% sure as 91% sure. 98% to 99% is 17
decibels to 20 decibels, which is less than 20% more evidence.

~~~
gwern
This paradox also helps illustrate why you might want to avoid percentages or
0-1 decimal probability in statistics - in some cases, the compression at the
end of the range can mask very important phenomenon. (0.01% and 1% look almost
the same as percentages or decimals, but can have different implications.)
Particularly important if you're doing anything at the tails of the
distribution, like thinking about how to increase extreme values (is
increasing the proportion of extreme-values from 0.01% to 0.10% extremely
important or utterly trivial?)

~~~
escherplex
Good observations. Look at how much easier it would be to solve this version
of the Martian potatoes problem which avoids percentages altogether:

Earth has developed an insatiable appetite for Martian potatoes with their
tasty pulp. Yuki runs a successful manufacturing plant on Mars which
synthesizes a product called 'Martian Instaspuds'; instant potatoes being more
cost effective to ship. Now these red potatoes (it's Mars after all) are
processed in lots of 100kg mass which consist of 1kg pulp to 99kg water,
1p:99w. For 'Instaspuds' this ratio must be reduced to 9p:1w. How much water
must be boiled-off in solar powered kilns on the Martian equatorial surface to
achieve this ratio?

1) posit the pulp mass remains constant at 1kg

2) x = end product mass; 9/10 x = 1kg; x = 10/9 kg; 1/9 kg = final water mass
in resulting mixture

3) subtract

------
mgalka
Cool concept. Took me a minute to think it through before I could make sense
of it.

I think it is the fact that they used potatoes that makes it counterintuitive.
Had it instead been a glass of water that had 1% of dissolved salt in it, it
would have been very straightforward.

------
JoshTriplett
The thing the article doesn't point out is _why_ it seems unintuitive.

If you phrased the question as "You have N pounds of potatoes", or with a
specific number _other_ than 100, it would come across as less unintuitive. As
you read, you see "100 lbs", and "99%", so percents and potato components are
both out of 100. So then you see 98%, which is 98/100...

~~~
vecter
Although it's somewhat true, if you had started with 14 pounds of potatoes
instead, I still think most people at first glance would expect the final
weight to be around 14, not half of it.

~~~
JoshTriplett
I agree, but I suspect fewer people would make the mistake. Partly because
more people would actually do the math, and partly because there's no
immediately attractive wrong answer involving "98/100" and "100 lbs". There's
still a somewhat attractive wrong answer of multiplying 98/100 by 14 lbs, but
there's one less reason to make that mistake.

This seems like the kind of thing that could be tested with a study. Ask a few
hundred people each version of the problem (ideally filtering out anyone who
has seen the problem before), and see how many get each version right.

(Potato paradox problem paradox: You ask 100 people to solve the potato
paradox. 99% of them get the answer wrong. You drop people who answered
incorrectly until you have a group where 98% got the answer wrong. How many
people are left?)

~~~
DanBC
Gerd Gigerenzer asks roughly similar questions of a wide range of people -
mostly medical professionals. It's scary how wrong people are, especially
considering that this is information they're using to medicate you or operate
on you.

Here's an example from his book:

[http://imgur.com/zO4zkl4](http://imgur.com/zO4zkl4)

Gerd Gigerenzer _Reckoning With Risk_.

------
amolgupta
q:100 people are seated in a room. 99% of them are enginners and 1% managers.
How many engineers should leave the room to make it 98% enginners and 2%
managers? a:50

~~~
mkagenius
Its better to leave the "and 2% managers" part to make it a little complicated
;)

------
FryHigh
Here is another variation.

A fresh lake gets infested with algae and the amount of algae doubles every
day. The algae covers the whole lake in 10 days. How many days did it take the
algae to cover half the lake?

~~~
pennaMan
9

------
DannoHung
I think it's interesting that _just_ the percentage stated makes it hard to
comprehend intuitively.

That is, restate the question with a different end water percentage and the
answer is immediately obvious:

> You have 100 lbs of Martian potatoes, which are 99 percent water by weight.
> You let them dehydrate until they're 50 percent water. How much do they
> weigh now?

And, of course, it's pretty easy to get "2 pounds", but your brain is pretty
fixed on the numbers all clustered together in the other example.

------
guelo
I don't think a simple algebra problem should be called a paradox.

~~~
jfreax
I think it's called a paradox (rightly), because it's not immediately
intuitive for most people.

~~~
andrewstuart2
Agreed.

> par·a·dox A statement or proposition that, despite sound (or apparently
> sound) reasoning from acceptable premises, leads to a conclusion that seems
> senseless, logically unacceptable, or self-contradictory.

------
dools
This strikes me as similar to the non-intuitiveness of compounding interest
(and related brain teasers like "bacteria doubling population every day" and
"pieces of gold doubling on a chess board").

------
antimora
Another intuitive way of thinking is to think in terms of proportionality
between water and potato matter. The weight of the potato matter remains
constant, and the amount of water can change, which in our case goes down. To
make the matter proportionally twice as bigger compared to water, one needs to
divide water by twice.

~~~
jeroen
Not entirely. You need to divide the total by 2, not the water. Dividing the
water by 2 would give 49.5 water vs 1 potato matter, for a total of 50.5. The
potato matter content would only be 1.98%.

------
andrewguenther
This equation models the amount of weight lost for various percentages of
change:

0.99 * 100 - (0.99-x)(100 - y) = y

This assumes that you are starting with 100 pounds of potatoes at 99% water
weight. Here's a WolframAlpha link:
[http://www.wolframalpha.com/input/?i=0.99+*+100+-+%280.99-x%...](http://www.wolframalpha.com/input/?i=0.99+*+100+-+%280.99-x%29%28100+-+y%29+%3D+y+from+0+to+1)

------
kwhitefoot
This is only a paradox because the framing of the question misleads people ,
partly by mentioning potatoes, into thinking that they are discussing
something that one might actually do in the way it is described. In reality if
you did this with real physical objects the part where it says: "You let them
dehydrate until they're 98 percent water" hides a process in which one of the
operations would involve picking up the potatoes. At this point you would
notice that they were much lighter than before.

So as stated it is a trick question, the kind of parlour game found in old
books of puzzles.

Finally, it seems to be a common failure of education to allow people to go
through their 'mathematical' training and leave them with the impression that
'percent' is some kind of dimension when in fact it is short hand for a ratio:
y is x percent of z. If z is not specified then you don't know what y is
regardless of how much effort you put into discussing x.

So I suspect that in real life there are not many occasions when the paradox
appears surprising.

------
thetruthseeker1
A better or more shocking way to understand how the numbers interplay is, to
reframe this like this. Imagine a vegetable mystical-vegetable, that had 99.9%
water and weighed 100 lbs. If I dehydrated it such that it now has 99% water,
what would be its weight? Yes, its 10 lbs.

------
vermooten
This is bs and you're all witches.

------
blahblah3
Let p be the percentage of water and m the total mass.

Then m(p) = 1/(1-p) . The "unintuitive" aspect is that we want to think of
this function as linear when it isn't.

By taylor's theorem, how bad a linear approximation to this function will be
is based on its higher order derivatives. We can get an idea of this by
looking at its second derivative:

m''(p) = 2/(1-p)^3. If you plot this graph, you'll see that it really starts
to blow up past 0.8, so the nonlinearities start dominating.

------
soup10
This is only confusing because of the potatoes.

If you said you had a pool that was 99% water, that changed to 98% water, the
massive weight drop would be much less surprising.

~~~
kzhahou
How is that clearer?? :-)

In fact, it's not even the same thing. A 100-gallon pool (for very small
people) that is 99% full has 99 gallons. 98% full has 98 gallons...

~~~
soup10
99% water by volume to 98% water by volume. or another way, sediments
increasing from 1% to 2% would make sense if half the water evaporated

------
chris_overseas
This is related to the base rate fallacy, the maths of which is frequently
misunderstood and can lead to very real consequences:

[http://metaist.com/blog/2010/02/disease-screening-base-
rate-...](http://metaist.com/blog/2010/02/disease-screening-base-rate-
fallacy.html)

------
jsingleton
Very good. I also like the birthday paradox.
[https://en.wikipedia.org/wiki/Birthday_problem](https://en.wikipedia.org/wiki/Birthday_problem)

------
powera
This isn't a paradox at all. It's a slightly non-intuitive result.

~~~
kzhahou
Since you're arguing terminology, I'll just quote the second dictionary
definition:

a seemingly absurd or self-contradictory statement or proposition that when
investigated or explained may prove to be well founded or true. "in a paradox,
he has discovered that stepping back from his job has increased the rewards he
gleans from it"

------
huhtenberg
[https://en.wikipedia.org/wiki/Potato_paradox#Explanation](https://en.wikipedia.org/wiki/Potato_paradox#Explanation)
\- oh my.

------
andrey-g
Took me an embarrassingly long time to figure this out. The key issue that
I've latched on to is that they've lost 99 - 98 = 1% of their water content,
which is false.

------
paulkon
Makes mathematical sense, yet not quite common sense.

------
golergka
But how is that unintuitive?

------
spacehome
I'd call it: "The Potato Algebra Problem"

------
taigeair
Any more of these interesting puzzles?

------
yummybear
Looks like I found my new diet.

~~~
will_work4tears
There's a guy (or maybe more than this guy) that has done just that:

[http://www.20potatoesaday.com/](http://www.20potatoesaday.com/)

------
Lejendary
Interesting

------
pervycreeper
Why is this a "paradox"? Not sure what this applies to either. It's not even
counterintuitive.

~~~
tjradcliffe
There is a certain type of mind that delights in paradoxes, to the extent that
even after an explanation has been clearly produced, the person will continue
to insist there is a paradox.

Naming things paradoxes attracts this sort of individual, so it's a kind of
marketing ploy for an idea: it won't just get it talked about, it'll get it
talked about _forever_ , because each generation of paradox-mongers will take
it up anew and discuss it and analyze it and do absolutely anything except
acknowledge that the specific flaw in the reasoning that leads to the
appearance of an impossible conclusion being true was exposed generations ago.

For example, consider the supposed paradox of the evening star and morning
star, which I will first state in the a way that makes it clear there is no
paradox, then restate in the traditional way.

"The Evening Star is Venus seen in the evening sky." "The Morning Star is
Venus seen in the morning sky." "Transitivity implies therefore that the
Evening Star is the Morning Star, since Venus seen in the evening sky is Venus
seen in the morning sky."

This is obviously stupid: no one would make this claim. The traditional
formulation actually depends on a falsehood, or at least on radically
incomplete statements:

"The Evening Star is the planet Venus." "The Morning Star is the planet
Venus." "Transitivity implies therefore that the Evening Star is the Morning
Star, since the planet Venus is the planet Venus."

At this point, you can spend millions of words explaining why the statements
identifying lights in the sky viewed at a particular time of day with a ball
of rock orbiting the sun are problematic. It will be pointless: paradox-
mongers will simply not let their nice toy be demolished.

Most traditional paradoxes have straightforward resolutions (frequently
involving inserting a knowing subject into them, like the person who observes
Venus) but none of them will ever be "solved" because of this purely
psychological resistance to the possibility of their solution on the part of a
very vocal sub-population.

One useful trick is to _never_ let a paradox be stated in the traditional way.
The first step in any discussion should be to restate the paradoxical
situation as completely as possible, usually by introducing the perspective of
particular individuals, as I've done above. Traditional paradoxes almost all
depend on very specific ways of stating them for their psychological effect,
and breaking out of that ritual pattern of restatement often makes them look
simply stupid. One can then ask what is missing in the ritual statement of the
paradox that makes it not seem stupid.

In the case of the Potato Paradox, the restatements we've seen here, which
introduce the ratio 1:99 as the way to think about the problem, is a good
example of this. Since the answer is not intuitive, the correct way to
introduce the problem is to make it intuitive, not to blurt out a non-
intuitive answer and expect the now-confused listener to catch up. That's just
bad pedagogy.

~~~
allendoerfer
The Potato Paradox seems to build on a misleading set-up and peoples
disability to picture logarithmic scales. The Venus Paradox seems more like
something a freshman says, who heard about Boolean algebra for the first time
and now applies it human language all the time.

------
iopq
That's not interesting at all. How do I downvote submissions?

