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I found a nearly perfect solution with 28 triangles for the square. Since the error is very small, this could be used for a jigsaw-puzzle. here are 3 versions with different locations of the flawed triangles:


2

To continue steven taschuk's suggestion : Let $N$ be some very big number, from which noone knows whether it is a prime number or not (for example the $33$-th Fermat-number). If $N$ is prime or composite, in both cases there is a proof for this statement, but the result is unknown. This can be applied to any decidable question whose answer is unknwon.


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It relates tu Collatz through the formula giving the elements of possible cycles in a Collatz sequence. Fi in (4*4*4*4 + 4*4*4*3 + 4*4*3*3 + 4*3*3*3 + 3*3*3*3) / (4*4*4*4*4 - 3*3*3*3*3) look at papers by Lagarias to find something in the direction.


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She is right - we already do generally assume the truth of the Riemann hypothesis, but knowing that RH is true does not provide any way to attack primes-based cryptography like RSA. Rather it's actually the opposite that is true, if anything. More on that in a second. But the implications of the Riemann Hypthesis (either way) for cryptography is greatly ...


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I agree with @avid19 that generally it isn't the results of the breakthrough solutions that matter as much as how they open up new ways of thinking and new classes of problems. Let's not forget that fields such as probability and combinatorics received great impetus by what today is a set of simple gambling problems addressed by Pascal and Fermat and ...


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Most of the time, the actual result isn't important as the theory. The reason why problems are unsolved is because either the math doesn't exist yet, or some connection between current fields has not been established yet. Either way, creating new math and connecting existing math are the real reasons why solving open problems is important. For example, ...


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Here's a theorem ($P=NP$) that would change the world if proven to be true: http://en.wikipedia.org/wiki/P_versus_NP_problem As it says on the wiki page: "Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game ...



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