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I need this kind of examples to motivate my friend to study mathematics. When he was a high school student, he is really good at math, but since he went to college (he majors in Math), he became unmotivated and lazy. He's always complained that higher mathematics is too abstract and therefore, not fun anymore. (This is partly because math textbook written in Vietnamese is so boring with monotonous theorem-proof-theorem-... pattern, without any interesting examples and problems). However, my friend is very enthusiastic about solving puzzles and elementary math problems. So, could you give me some examples of puzzles that require some deep math to solve, which illustrate my belief that "clever and quick is not enough, deep is more important"?. I hope I can make him stop showing off and start studying math seriously.

Some good examples I know are:

  1. Instant Insanity puzzle
  2. Sam Loyd's 15-puzzle
  3. Lights out Puzzle (This puzzle can be solved by trial-and-error, but to know which configuration is solvable requires linear algebra).

I do not ask for seemingly elementary problems that turn out to be open for a long time (like Fermat's last theorem). I need examples that I can explain the solution to him.

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  • $\begingroup$ Could you also edit your question and add links to the puzzles you have mentioned ? $\endgroup$ – Shailesh Aug 14 '16 at 12:15
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    $\begingroup$ If your friend finds the "theorem -- proof -- theorem ..." style monotonous and higher math too abstract then my best advice for him is to change his major. $\endgroup$ – Gregory Grant Aug 14 '16 at 12:18
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    $\begingroup$ Yeah, but the problem is that the textbook we use does not contain any good examples and the exercises are always about checking if you understood the material (like, prove a map is continuous). Even worse, he cannot read books written in English. I know that he just prefers to think in a concrete way. Anyway, it is too soon to advise him to change his major. $\endgroup$ – Pluviophile Aug 14 '16 at 12:31
  • $\begingroup$ I like your nickname. Have you seen the musical film "Singing in the Rain" with Gene Kelly? $\endgroup$ – Piquito Aug 14 '16 at 12:53
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    $\begingroup$ Introducing him to some practical uses of math (such as solving DE's and numeric's) might do loots of good for him! $\endgroup$ – Piotr Benedysiuk Aug 14 '16 at 13:06
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Jordan's theorem is not really a puzzle but some believe that it is when they first learn about it. A Jordan curve is a non-self-intersecting continuous loop in the plane (see the figure). This theorem states that the plane is divided by all Jordan curve into two non-intersecting regions, one inside and the other outside.

A proof "without errors" of Jordan's theorem requires a certain non- elementary level of mathematics.

enter image description here

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  • $\begingroup$ Thank you very much! I can think of a related example is Lakes of Wada $\endgroup$ – Pluviophile Aug 14 '16 at 13:06
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    $\begingroup$ With the difference that Jordan's theorem is "obvious" (do not forget the quotes!) $\endgroup$ – Piquito Aug 14 '16 at 13:34

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