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Title might be a bit vague, so I will explain further here.

I am compiling a list of examples of how a person may realize some mathematical result is either obvious or unsurprising from the understanding of some high-level math concept or abstraction, but in a way that the intuition is potentially straightforward and simple enough that there is no need to work out the details on paper or computer. Of course, this is subject to interpretation, but anything even tangential to this I would appreciate.

I am looking for several categories of examples including but not limited to:

  • A clever solution to a difficult problem
  • Philosophical or practical understanding of some concept that applies to real life
  • "Surprising" mathematical facts understood very clearly (or just not so surprising) once a concept is understood

Here are some examples of what I'm looking for:

Brouwer fixed point theorem: If one understands the intermediate value theorem and Brouwer fixed point theorem, then the wobbly-table problem becomes trivial: Suppose you have a table with 4 legs and a surface that is uneven. You can stabilize the table on the uneven surface by simply rotating the table until it is stable. By the Brouwer fixed point theorem, that stable configuration must exist.

Partial differential equations: "In order to eat as much as possible in a day, one should not eat as much as possible all day." If someone wanted to maximize their food consumption, rather than continuously consume food the entire day, it may be optimal to consume 3 large meals or 5 small meals instead. This is understood through partial differential equations: the rate of digestion may be dependent on various other factors such as the amount of food in the stomach or appetite. This may be a concept that is already considered common sense, but I still accept these because it allows these ideas to be understood a bit more technically.

Coordination games (game theory): A coordination game in game theory can help us understand how cultural differences manifest. "Is burping considered polite or rude?" is a question that can be simulated by a coordination game and understood to be dependent on locality. This helps us understand the idea that someone with different cultural values isn't "wrong" but simply grew up in an environment which has stabilized at a different Nash equilibrium. Once again, this is very much common sense to a lot of people, but it is understood better technically through the understanding of game theory.

Function asymptotic growth: (3 examples) The wheat and chessboard problem is very famous, and is a go-to example educators use to demonstrate the monstrous growth of exponential functions. However, we can further grok that growth by understanding the growth rate of functions and comparing functions. In this particular case, this result is "obvious" once we realize that we [can] measure the amount of rice grains by volume, which grows at a rate of $f(x)\sim x^3$ but the number of rice grains grows at $f(x) \sim 2^n$.

Additionally, other facts become similarly "obvious" or "unsurprising" with a little bit of additional mathematical background.

For instance, the fact that there are more possible games of chess than there are atoms in the observable universe is not very surprising once it is understood that counting functions are often times exponential or super-exponential, and the number of atoms is proportional to the volume (which is once again $f(x) \sim x^3$).

The fact that all of the digits of the number $$9^{9^{9^9}}$$ cannot possibly be contained within the observable universe is also no longer surprising.

Topology (genus): A spherical planet's surface cannot be simulated properly by taking a rectangular or square map and simply allowing it to "wrap" around (connecting the top edge of the map with the bottom edge, and the left edge with the right edge). This "wrapping" results in a torus (genus $1$), which is topologically distinct from a sphere (genus $0$).

Infinite ordinals: Goodstein sequences, when evaluated naively, seem like they would not only grow extremely quickly, but grow forever. A very basic understanding of infinite ordinals is enough to directly map the sequence to an ordinal sequence, making the fact that the sequence must eventually terminate to $0$ quite unsurprising and obvious.

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    $\begingroup$ Not exactly deep math, but: lossless compression on arbitrary data is impossible (by the Pigeonhole Principle). $\endgroup$ Apr 29, 2020 at 18:46
  • $\begingroup$ The concept of known unknowns and things like Chaitin's constant. $\endgroup$
    – rtybase
    Apr 29, 2020 at 20:31
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    $\begingroup$ Or impossibility to measure at sub Planck length level makes some mechanical systems fundamentally unpredictable, also explained here. $\endgroup$
    – rtybase
    Apr 29, 2020 at 20:34

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Enumerate the rationals. Put an interval of length $\frac{1}{2^n}$ around the $n^{th}$ rational. Can every real number be covered? Since the rationals are dense in the reals, one might think "yes", but basic measure theory says that the answer is definitely "no".

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  • $\begingroup$ And here......... $\endgroup$
    – nonuser
    Apr 30, 2020 at 19:50
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How many humans will ever live? Doomsday Argument attempts to predict the number of future human species using basic probability and few assumptions.

Can a finite universe have no edge? Yes. Consider a flatlander on sphere walking straight. She will eventually go back to point from when she started. Same can happen in more than two dimensions - in Three-torus, for example.

Surface to volume ratio and heat loss. Bigger animals have a lower surface to volume ratio. This contributes to a faster heat loss. Based on that we can make a rough prediction that natural selection would favour bigger animals in cold regions (all else unchanged) and vice versa (see: Gigantothermy, Bergmann's rule).

Use of graphs in social science. Some biases become obvious when we look at the graphs. For example Majority Illusion (see: ncase - crowds for interactive explanation).

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A list of possible examples.

Ancient problems. Squaring the circle. Trisecting the angle Doubling the cube. The quest for the proof of the fifth Euclidean postulate and its final solution by models of non-Euclidean geometries.

Combinatorial geometry. Borsuk’s conjecture. Hadwiger-Nelson problem.

Combinatorics. Ramsey’s theorem (finite and countable infinite versions). The latter implies that each sequence of real numbers has a monotonic subsequence. Van der Waerden’s theorem.

Geometric topology. Möbius band (an one-sided surface) Klein bottle. Drum theorem. Ham sandwich theorem. Borsuk–Ulam theorem. Jordan curve theorem, Jordan-Schoenflies theorem, and Alexander horned sphere. Poincaré–Miranda theorem.

Graph theory. Non-planarity of graphs $K_5$ and $K_{3_3}$. Four-color theorem.

Logic. Gödel incompleteness theorem based on the liar paradox.

Number theory. Euclidean proof that there are infinitely many prime numbers. Bertrand’s postulate. Asymtotics $x/\log x$ for the prime counting function. Fermat's last theorem (see, especially a book with the same name by Simon Singh).

Set theory. The diagonal proof that the set of real numbers is uncountable. A proof that the cardinality of the segment equals the cardinality of the square. Banach-Tarski theorem.

Social choice theory. Arrow's impossibility theorem.

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  • $\begingroup$ What is the "easy to understand" conclusion of the two linked combinatorial geometry examples? I don't see the connection between the theorems and such a concept, could you explain? $\endgroup$
    – zhuli
    Mar 24, 2021 at 17:20
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Well, understanding inverse of a function may be advanced for a few people, so this is what I have come up with.

Suppose you have all $13$ cards of any suite from a deck of playing cards. Your challenge is to assemble the cards in such a way that if you repeat the process "displace the card from the top and place it at its bottom, and remove the next card and and place it face up" enough number of times, the cards must be face up in increasing order of their value. One can obtain the assembly via trial and error in some time, which is actually : $$7,A,Q,2,8,3,J,4,9,5,K,6,10$$ But this assembly can be obtained much more easily by using "inverse of a function".

Basically, our function here is "displace the card from the top and place it at its bottom, and remove the next card and and place it face up" whose inverse will be "pick the topmost card facing up and place it on the pile, and remove the bottommost card from the pile and place it at the top". Employing this inverse enough number of times will generate the required assembly.

Also, the concept/intuition of invertible function can be understood by saying that the above function was invertible but the function of "shuffle the cards at RANDOM" is not invertible due the the word "random".

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