# What are some examples of a mathematical result being counterintuitive?

As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.

My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite volume but infinite surface area (I later learned of Koch's snowflake which is a 1d analog). I just remember doing out the integrals for it and thinking that it was unreal. I later heard the remark that you can fill it with paint, but you can't paint it, which blew my mind.

Also, philosophically/psychologically speaking, why does this happen? It seems that our intuition often guides us and is often correct for "finite" things, but when things become "infinite" our intuition flat-out fails.

• Why does it happen? Because our intuition is developed by dealing with finite things: it is quite unsurprising that we are surprised by phenomena specific to infinite objects! This is exactly the same as the fact that our bodies are trained to move and act under the effect of gravity, so when we are in space we become clumsy and need to retrain. Intuition is not fixed: if you study phenomena associated to infinite objects, you develop an intuition for that, and presumably people working with large cardinals, (cont.) – Mariano Suárez-Álvarez May 2 '12 at 1:13
• (cont) or strange objects like graphs with chromatic number $\aleph_8$ or Banach-Tarski partitions of a sphere, after a while find them just as intuitive as you and me find the formula for the area of a triangle. Intuition is, in most situations, just a name we put on familiarity. – Mariano Suárez-Álvarez May 2 '12 at 1:15
• Philosophically / psychologically speaking, human brains weren't adapted for intuiting mathematical truths. The fact that we can repurpose our brains to do mathematics at all (beyond counting etc.) is astonishing. As for Gabriel's horn, I don't think this is a good example: see math.stackexchange.com/a/14634/232 . – Qiaochu Yuan May 2 '12 at 1:20
• I think remarks like "you can fill it with paint, but you can't paint it" are actually not helpful. In trying to appeal to our everyday intuition, they get in the way of mathematical understanding. Of course, you can't paint Gabriel's Horn (it's surface area is infinite) but you can't fill it with paint either (because paint molecules have a finite size, and Gabriel's Horn gets infinitely thin). Or, more prosaically, you can't fill Gabriel's Horn with paint because it's a mathematical idealisation that doesn't exist in the physical world. – Chris Taylor May 2 '12 at 7:35
• "In mathematics you don't understand things. You just get used to them." ---John von Neumann. – Nate Eldredge May 2 '12 at 19:33

Another elementary one. There is a configuration of 30 convex bodies in 3-dimensional space with disjoint interiors that "cannot be taken apart with two hands". That is, it's impossible to split up the set of bodies into two nonempty subsets and, by a rigid motion, move one of the subsets away to infinity without disturbing a member of the second subset. See http://www.cs.ubc.ca/nest/imager/contributions/snoeyink/sculpt/theorem.html and http://www.springerlink.com/content/v32326564w8l4n75/

Although Dan Brumleve and plm linked the list of paradoxes, the famous Braess's paradox deserves a special attention.

"For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times."

This, is the most counterintuitive fact that I ever saw:

Blue Eyes Islanders Question: http://www.math.ucla.edu/~tao/blue.html

This question was created by mathematician Terence Tao.

• And on the 101st day everyone else commits suicide. Tragically – Thomas Ahle May 25 '13 at 9:53
• Not really, @Thomas, because no islander has reason to believe that their eye color is exactly one of blue or brown. – superAnnoyingUser Jan 6 '14 at 0:26
• Yes student, only blue eyed islanders commits suicide. – Gaston Burrull Jan 11 '14 at 5:05
• I am unsure that this question should be attributed to Terry Tao. I don't know who it should be attributed to, but I have had a quick search about and I would be surprised if it was actually due to him... – user1729 Apr 22 '14 at 13:40
• I don't really find this one counterintuitive. – Michael Hardy May 2 '12 at 17:08
• Probably the guys who scratch their head at Zeno's paradox are also the ones who are stumped by this one... – J. M. is a poor mathematician May 2 '12 at 17:27
• @MichaelHardy we now just proved that intuition differs between different people. I thinnk that one who knows and hence "feels" some knowledge area better, will see things as basic and intuitive, while people new to it, will find same ideas as confusing and counterintutive. Intuition adjusts itself to our knowledge. – Sandman4 May 2 '12 at 17:34
• I don't see why it is offensive, I must say. Both scenarios rest on there being such a thing as the geometric series, for starters. – J. M. is a poor mathematician May 2 '12 at 17:38
• It's offensive because I felt offended and it's counterintuitive because It's against my intuition. The common between the two (feelings and intuition) is that both are personal and not a scientific concepts. – Sandman4 May 2 '12 at 17:42

For quite a while, when I was very young, I thought that the solution to the famed brachistochrone problem (which asks about the shape of a wire where a bead sliding frictionlessly on it under the influence of gravity will take the least amount of time to finish) was not a straight line was pretty counterintuitive, after being repeatedly told that the straight line is the shortest path between two points. (To spoil: the true brachistochrone is an inverted cycloid, which is the curve traced by a fixed point on the rim of a rolling circle.) I went through the derivation a number of times before finally being convinced.

I find it counterintuitive that the rational numbers have zero measure. This means that for any $\epsilon$, no matter how small, we can find a collection of intervals that includes every rational but whose total length is less than $\epsilon$.

Such a covering includes all the rationals, but somehow must omit nearly all the irrationals. The question has some up here several times about what irrationals are missed (1 2 3), and Asaf Kargila recently described the result as “baffling”. So I am not the only person who is surprised by this.

Von Neumann famously said that in mathematics you don't understand things, you just get used to them. This for me is one of those things.

(The countability of the rationals, or the uncountability of the irrationals, may be similarly baffling, but I have gotten so used to both that I no longer find either one baffling, and I am not sure which one should be considered counterintuitive.)

Seat $n$ dining people at a round table. Is it possible to re-seat these people $n-2$ times during that dinner so that at the end everybody has been seated to the left of everybody else?

Intuition suggests that this works (at least) for $n$ prime. I find it highly counterintuitive that this is possible if and only if $n \not\in \{4,6\}$. This is a special case of the Oberwolfach problem, see also this answer by Ben.

It's fairly straightforward to prove that the Cantor set is uncountably infinite (i.e. it has the same cardinality as the real numbers). You would think that such a set would have at least positive Lebesgue measure. WRONG-IT ACTUALLY HAS MEASURE ZERO! This blew my mind when I first learned about in undergraduate real analysis and it's why it puzzles me why Paul Halmos famously described measure theory as "a generalized kind of counting". This example seems to show that measure has nothing to do with cardinality, which is what most of us think of as counting in abstract sets!

• The standard Lebesgue measure definitely has something to do with cardinality. If a set has positive measure (or is unmeasurable) it has the same cardinality as the reals. Measure is a more refined sense of counting-so refined, in fact, that it is only really interesting on a class of sets all of which have the same cardinality. – Logan M May 4 '12 at 7:56
• The existence of "counting measure" on any set seems to give one indication that measure theory is a generalized kind of counting. – Pete L. Clark May 5 '12 at 17:52
• @Pete: I think he was trying to ping you. – The Chaz 2.0 May 8 '12 at 4:00
• If instead of taking away the middle 1/3 at each iteration, you instead take away the middle 1/4th, you end up with a "fat" Cantor set which, surprisingly, has positive measure. – Nick Alger May 16 '12 at 2:07
• @LoganMaingi: Solovay showed that it is independent of ZFC whether there are nonmeasurable sets with cardinality less than $2^{\aleph_0}$. See this answer of JDH for a survey of related results. – Jonas Meyer May 17 '12 at 3:40

I always found the main correspondence of class field theory counter-intuitive. Somehow the (Abelian) field extensions of a number field $K$ correspond to arithmetic objects in $K$ itself.

You wouldn't think that the possible ways of extending a field (albeit in a nice way) would just depend on the field you started with.

• "You wouldn't think that the possible ways of extending a field (albeit in a nice way) would just depend on the field you started with." What would it depend on, if not on the field you started with? – Phira May 6 '12 at 10:04
• Maybe my wording is not clear, I mean "completely dependent on". Intuitively a field extension comes from adding in something not in your original field. This could be anything. – fretty May 6 '12 at 10:17

I am surprised nobody has given this example: It was once thought that continuous functions have derivatives at almost all points (as other answerers have used). Then comes the Brownian motion model that generates a random curve, which, with probability 1, is a curve without derivative at any points!

Independence in Statistics. If I have a box with red & black socks, and I take one, and then another one the probability of both socks being red is sometimes different from the case when I, instead, pick a pair. (EX.If all socks are connected to every other with a string, like a complete graph, and then I pick some string, automatically choosing the socks at either end as the pair selected).

Set theory dealing with infinite sets is full of counterintuitive stuff, but comparing Cantor's diagonal argument (which proves [0,1] is uncountable; and is indicative of a common technique for showing the uncountability of an infinite set) to the ordinal numbers and the fact that there is a first uncountable ordinal (provable without the axiom of choice) is profoundly bewildering, even to someone well versed in both constructions.

The St. Petersburg's Paradox was counter-intuitive enough for me to change how I think of $$\infty$$.

Suppose you start with \$1 and a coin flip. If the coin lands tails, you take all your winnings (\$1 at this point) and the game ends. If it is heads, you get you double your winnings and get to continue flipping. Each heads doubles your winnings until you get a tails which ends the game. Question: what's the average winnings from this game?

The answer is (you guessed it since I spoiled it) $$\infty$$. It's quite easy to show:

$$\mathbb{E}[\text{winnings}] = \sum_{k = 1}^{ \infty } (1/2)^k 2^{k-1} = \infty$$

But notice that with probability 1, you will always get a finite amount! So how is the average value of something that is finite with probability 1 infinite?!?

If you think of $$\infty$$ as number, this seems to make no sense. The average of a set of values cannot be larger than all of the values. But if you think of $$\infty$$ as simply meaning "without an upper limit", it does seem to make sense. That is, if one were to set a fixed price to play this game, for any fixed (and thus finite) price, if this game is played enough, the average return will exceed that finite price.

## protected by J. M. is a poor mathematicianMay 6 '12 at 4:09

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).