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"Simple" problems

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Zhen Lin

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The aim of the game: Write a deceptively simple question or problem.

The catch: A solution, or a link to further information, must be provided for the desperately curious.

What is 1? What is 2? What is 3? Define the natural numbers. Demonstrate that 2 + 2 = 4 using your definition.

Mathematicians have struggled to define numbers in a useful, consistent manner for a long time. One such system is Peano axioms.
 
Why is it that for natural numbers [1, 2, 3, ...] a, b, c and n, and for n greater than 2, are there no solutions for the equation an + bn = cn?

This problem is [wp]Fermat's last theorem[/wp]. It is a statement so simple that even a grasp of high-school mathematics is sufficient to understand it; and for its simplicity, it had gone unsolved for nearly 3-and-a-half centuries.

Mathematicians chipped away at the problem, proving it for specific values or categories of n, but the fact that there are an infinite number of n made such a strategy ultimately infeasible. Finally, in 1995, a proof by [wp]Andrew Wiles[/wp], with the assistance of Richard Taylor, was published, and the conjecture was proven true once and for all.
 
Is it true that every even number greater than 2 can be written as the sum of two primes?

This is Goldbach's Conjecture, one of the oldest unsolved problems of number theory. Like Fermat's Last Theorem, it is simple enough that anyone with basic mathematical education can understand it, yet it has gone unsolved for over two and a half centuries. It has been shown to be true for all even numbers up to somewhere around 10^17, but there is no general proof at this time.
 
Why are there no living animals as large as the dinosaurs?

There aren't any structural reasons why terrestrial mammals couldn't be as large as the dinosaurs. There are a few factors that come into play including length of gestation, food supply, and the effects of gravity.
 
What is the most straightforward way to define ignorance?

I don't know.

Spoiler tags don't work, so you might have to wait for the answer.
 
If a tree falls down and no one is there to hear it, does it make a sound?

I have no idea!
 
But seriously, If a tree falls in the woods and no one is around to hear it, does it make a noise?

Sound is merely vibrations in the air which only becomes noise if it makes the bones in the ear vibrate. So if no ears are around, then effectively, no noise is made.
 
Here's a riddle for you: A girl is facetious. She is also abstemious. She gets pneumonia. Based on these clues, what American tree is she most like?

The Sequoia. She only likes words with all 5 vowels in them. Sequoia is the only tree name that contains all 5 vowels.
 
Those are not so much "simple" problems as they are out-of-the-box-thinking riddles...

Anyway.

Imagine you are on a game show. There are three doors. Behind one door is a goat. Behind another is a car. The other door has nothing behind it. You pick one. The host must open one of the two doors without a car behind it, and he does so. He offers you a chance to switch your choice to the other door.

Why do your chances of winning the car double when you switch?

This is called the [wp]Monty Hall problem[/wp]. Several explanations and proofs are listed in the linked article.
 
FabuVinny said:
But seriously, If a tree falls in the woods and no one is around to hear it, does it make a noise?

Sound is merely vibrations in the air which only becomes noise if it makes the bones in the ear vibrate. So if no ears are around, then effectively, no noise is made.
Click to expand...

Hah???? Your just confusing me now. Sound is sound you can hear it.

What is the most spoken language in the world?
I think its Either English, Spanish or Chinese but theres a great chance that its English.
 
That's not a deceptively simple question.

No.

Yes, it is a very deceptively simple question. It really depends on your definition of 'language', and 'speak'. If by speak you include second, third, fourth, nth-language speakers, then, indeed, it's probably English. By first language speakers, depending on your definition of language, it's Chinese, or Mandarin.
 
Even if its the first language speakers English would win because if you put together all the English countries like UK, America, Australia, Canada- its more than the people who speaks Chinese or mandarin.

Whats π (Pie)=?
 
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Whats π (Pie)=?

"Pie," as you've spelled it, is a dessert made with fruit. It's eaten best with ice cream.

"Pi," the number, is 3.14159265359...ad infinitum.

I have a question: what is the difference between a proof, a conjecture, and a lemma?
 
Barb said:
Whats π (Pie)=?

"Pie," as you've spelled it, is a dessert made with fruit. It's eaten best with ice cream.

"Pi," the number, is 3.14159265359...ad infinitum.

I have a question: what is the difference between a proof, a conjecture, and a lemma?
Click to expand...
A conjecture is a mathematical statement that is believed to be true, and has withstood substantial testing, but has not been proven. When proven, it becomes a theorem, which can then be used as a given in later work. A lemma is a theorem with little significance in and of itself, but which is useful in proving larger results. A proof is simply a set of statements which lead to the logical conclusion that a given statement must be true.

And pi, the ratio of the circumference of a circle to its diameter, is equal to 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - ...), among other formulas, and is approximately 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384...
 
Pooh, I only know it as 3.14159265358979... Perhaps I refer to my phone number as a substring of the decimal expansion of pi. Ah, yes, there it is, starting at the 14569419th digit after the decimal point. (14159... starts at the 1st digit after the decimal point.)

Perhaps Murgatroyd's phone number is something like 314-1593 or 278-1828.

Here's a "simple" question: is every possible string of digits equally likely to occur in the decimal expansion of pi as it would be in a randomly generated sequence of numbers?

Another "simple" question: is it possible for pi in some exotic geometry to be less, or more than 3.14...?

john1991 said:
Even if its the first language speakers English would win because if you put together all the English countries like UK, America, Australia, Canada- its more than the people who speaks Chinese or mandarin.
Click to expand...

No.

There are some 700 - 800 million native speakers of Mandarin (depending on survey), and some 1,080 million native speakers of any sort of Chinese. There are 1.3 billion ethnic Chinese, though I will grant that a good portion, if not most of the overseas Chinese population (about 34 million) do not speak any sort of Chinese as their first language, and some of them don't even speak any sort of Chinese.

The sum of the populations of the US, UK, Australia and Canada is only about 410 million. Other countries, which have English as one of their official languages (interestingly, USA does not), do not necessarily have English native speakers as 100% population, or even more than 50%...
 
Perhaps Murgatroyd's phone number is something like 314-1593 or 278-1828.
Click to expand...

If Murgatroyd's phone number contained the digits '314' anywhere, I think I would fall off my chair laughing.

Here's a "simple" question: is every possible string of digits equally likely to occur in the decimal expansion of pi as it would be in a randomly generated sequence of numbers?
Click to expand...

I don't know, but you might find this interesting reading. One of the funniest quotes from that article is: "If you want truly unpredictable, unrecreatable, random numbers - let my wife balance your checkbook."

But then I came across this: "Well, there is a reason why mathematicians consider that statistics are not a branch of mathematics." It isn't?

[EDIT: I'm sorry if I've used up my quota of dumb questions for the day, but I'm not a mathematician and I'm trying to wrap my brain around some of the posts in this thread.]
 
Barb said:
If Murgatroyd's phone number contained the digits '314' anywhere, I think I would fall off my chair laughing.
Click to expand...
It doesn't, but for my first two years of college, my dorm room number was 314.
 
Heres a question?

How can you fit 9 horses in 7 stables?

Its a kind of a riddle!
 
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