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Main Answer I have a PhD in Mathematics and came across this question. To be honest, I dislike almost every single answer, except maybe L.Dutch's answer concerning Wile's proof of Fermat's last theorem. However, I do think there is a much, much better candidate, and one that would make every mathematician reading your story quite delighted: https://en....


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Dogs. https://www.navysealmuseum.org/raven-nsm-multipurpose-canine If the problem is the limitation of human senses, dogs are the answer. Dogs have gone to war since there were dogs, and commandos and dogs work well together. Illusions designed to trick humans will be obvious fakes to the dogs. They will smell right through disguises of any sort. The ...


72

The Riemann hypothesis is what you're looking for. Basically everyone in number theory assumes it to be true (although no one can prove it). Variants of it have been proven in other settings. Many results, including entire theories of math, are conditional on its truth; these would all collapse if it was shown to be false. The discovery of even one ...


61

Date of the next lunar/solar eclipse. See “Antikythera mechanism”: https://en.m.wikipedia.org/wiki/Antikythera_mechanism Eclipses follow such a complex pattern, they cannot be estimated from previous events, but as the Greek mechanism shows, they could be calculated.


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Since you're interested in something currently unsolved, you don't really care about how protagonist will solve it, only that he does so, let's go for something extremely simple, simple enough that even a reader with only basics of mathematics can understand: Collatz conjecture (also known as 3n + 1 Conjecture) "Consider the following operation on an ...


52

As far as you want. Humans have trouble conceptualising large numbers. We can’t count them, so above a certain point we see the number ‘178654’ and our brains turn it into ‘many’. Doesn’t change the value of the number, just how we intuit about it. For anything bigger than the number we can imagine (varies from person to person) we start doing maths rather ...


47

No such thing exists. All mathematics is a type of language. Like language, it looks mysterious to people who don't speak it. But if you study it enough, you will understand it. There are no exceptions. (*) Calculus was once an arcane branch of knowledge known only to Newton, Leibniz, and their handful of peers. It made them gods in terms of their ability ...


46

The biggest visual clue that you're on a much larger planet than Earth is a mismatch between atmospheric haze (related to distance), perspective, and horizon distance. A ship out at sea, for instance, would have much more perspective "shrinking" and haze coverage while still above the horizon than you're used to. If you have any ability to measure distance,...


44

The big clue is that you'll never see something disappear over the horizon. On an Earth-sized planet, the horizon is about 5 km away. Under ideal atmospheric conditions, you can see things up to 300 km away. Seeing something drop below the horizon is no problem. On a Sun-sized planet, the horizon is now 550 km away, but atmospheric conditions are no ...


37

Probably not While it is true that a Base 8 system would mean that the general public would be able to work with programming numbers and hexadecimal better, anyone who uses those number systems with any frequency very quickly acclimates to the various base systems within any programming language. There are programmers who can just offhand convert numbers to ...


34

Your password system won't work on a flanking enemy. They will kill you why you are asking the password. IR visors will tell you where there are objects warmer or colder than the background. Those happen to be humans. Make sure that your team wears a clearly identifiable mark visible in the IR, so that if they see sergeant John Doe without the mark they ...


31

This definitely wouldn't work in pure math, as there's nothing so "destructive" you could prove that wouldn't make you a celebrity among mathematicians. Finding a contradiction in ZFC would get you a Fields medal. Disproving the Riemann hypothesis would get you a Fields medal and a million dollars. There would never be any question about whether to publish, ...


28

Instead of looking for new math that the children can do, show them learning math faster. All new math is built on the old. To do some incredibly complex proof, you'll typically need algebra, equations, maybe calculus or group theory or probability or what-have-you. The point is, it will be clear that these children are exceptional WELL before they invent ...


27

Let's assume that this student wants to begin by understanding the twin pillars of modern physics: quantum mechanics and general relativity. There are several major tools in the toolkit of anyone studying both of these theories at a basic level: Calculus (single-variable and multivariable) Differentiation Integration Operators such as divergence, gradient, ...


27

Others have mentioned some famous conjectures such as the Collatz conjecture and P = NP, but I think it's awfully unlikely that a freshman math student would be able to solve such a problem. About the Collatz conjecture, Paul Erdős famously said that "Mathematics may not be ready for such problems"; and about P = NP, Scott Aaronson wrote that "any proof will ...


27

The average volume of an adult human is about 62 liters. Assuming that the aliens don't use any process that causes the loss of volatiles elements, the only saving from liquefying a human body would come from the air volume taken by the lungs and bowels. In an average adult the lungs account for about 6 liters of air, while the bowels I wasn't able to ...


22

Since we know that the density of a human is very close to 1kg/l (we float in water, but only just), a human’s volume in litres is pretty much the same as their weight in kilogrammes.


21

Estimate exponential growth using grains of rice and a chessboard. The story goes: The ruler or India was so pleased with one of his palace wise men, who had invented the game of chess, that he offered this wise man a reward of his own choosing and he said to the man: “Name your reward!” The man responded: “Oh emperor, my wishes are simple. I ...


20

Goldbach's conjecture states that every even integer greater than two can be written as a sum of two primes. If one could find a counterexample the problem would be solved (although currently all candidates smaller than the order of 10^18 have been tried). Alternatively if one could give a formula the problem could also be solved.


20

How good is liquefaction? L.Dutch's answer is the right concept, but his numbers are wrong. 6 liters is the maximum inhalation of an average adult male; however, men have much larger capacity than women and normal respiration does not fully inflate the lungs. The 62 liter volume of an average adult assumes a resting inhalation volume which actually ...


19

A perfect number is a positive integer N such that N is the sum of its divisors (other than itself). For example, 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 Question: Does there exist an odd perfect number?


18

Where a cannon ball will land. A problem of extreme importance in the Middle Ages, why Newton and Galileo were studying gravity, and (because of the scale involved) nearly impossible to estimate with the precision desired by commanders. Because of the great mass of a cannon ball, the effect of wind and complicating factors of air resistance are ...


18

The following statement is unproven but widely assumed to be false: e+π is rational. A rational number is a number you can write as a fraction. For instance "0.25" is rational because it can be written as "1/4". "7" can be written as "7/1", "14/2" or whatever. We are absolutely sure that π and e are not rational, but mathematicians only assume that this ...


18

Back in the day, you didn't specify how many bricks and logs it would require to build a house. You just baked bricks and made logs while building until the house was finished. Any leftovers were kept for repairs or the next time a house was built: "We need to build 1 house, and we will need many bricks and logs and buckets of mortar - until we need 0 more.";...


17

A very well known conjecture (hopefully anybody with a CS degree would recall this, for example) is that there does not exist an efficient algorithm to take the discrete logarithm, in the most general case. This fact is sometimes used in cryptopgraphy. There are enough cases in which it is computationally tractable that it is plausible, at least, that your ...


16

For basic physics, you need (multivariable) calculus and linear algebra. This is basic literacy. You won't get anywhere in physics without them. There are some differential equations too, but one tends to learn that on a case-by-case basis as one studies examples. For general relativity, you need Riemannian geometry. Talking about curved spacetime only ...


16

As a computer programmer and electronic engineer I can say the answer is no. For a fairly short period in our history there was some advantage to being familiar with base 8. When computers were new and not very powerful and more importantly when software development often involved understanding the internal workings there was an advantage to base 8 because ...


15

I find it hard to believe that a proof or disproof of any mathematical statement could cause "many careers in tatters". If professional mathematicians around the world all made the same mistake, it hardly reflects poorly on any individual mathematician. Moreover, when a mathematician makes a significant mistake, it almost always is for some non-...


14

Inventing a new technology might be impossible, but I didn't see any restriction about purchasing existing technology. That being the case: Laser Imaging / LIDAR There currently exist handheld 3D laser-imaging devices which can be used continuously while in motion. If each member of the team (or even just a few) is equipped with such a device, they'd ...


14

They'd develop the same mathematics we did. they have developed complex spoken and written languages There is no rational reason a species capable of developing complex spoken and (particularly) written languages will not develop equally complex written mathematics. It's a natural progression. Counting predates human written history. We have no idea ...


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