Math Is Fun Forum

^ Agreed

II 64.

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Yes, that is it! Excellent
The only distance that is not given is AE = PR, but I'm sure it is reasonably well-known that 1-1-√2 is a Pythagorean triple since the hypotenuse is √(1^2+1^2) = √2.
Is there any restriction on the flight of the flies? Because if not, they can just fly from A directly to C. That distance has to be √2 as well, or about 1.4142, since it is completely symmetric to the distance AE. Because there is so much information provided, you can prove this with any rule you like (drawing AC on the left gives two triangles, each with 90, 45, 45 degree angles and two sides of length 1).

The ant, since that is the simpler, will prefer to walk AP + PC. The length PC is √5 / 2, since that is √(1^2+(1/2)^2), which is about 1.1180. Therefore AP + PC = 1/2 + √5 / 2 which is about 1.6180.
You can prove this simply by asserting that PC is the shortest distance from P to C and that PC + AP must be shorter than the sum of AE or AR and any available route (in fact, it is shorter than even AE + ER or AR + anything else (AR is 1.5)).

The termite is more complicated. The problem is to optimise the distance AX + XC where X is some point on PR. It's possible that this cannot be less than 1/2 + √5 / 2, and the termite will go the same way as the ant, but I haven't proved it yet.
AX = √(2x^2 + 1/4) where x is the fraction of the distance PR traveled from P (in other words, x = PX/PR).

I am very ignorant of higher maths and will be unable to shed much light on this problem, but just to get the ball rolling I think it's clear that it must be more than 12 (since there is greater than a 50% chance of winning the first 6 and that guarantees a minimum 12).

Hi;

2(1/2 + 3) - 6(1/2 - 1/3)

Hi!

Although I never actually bothered to learn the short-cut method with frequencies ;s
Median is 70. Standard deviation is √(345/11) / 2 ~ 2.8002

Although I think it is mathematically possible that we are in some sense crossing infinity when we walk across a room, or even situated within an infinite cosmos, philosophically I find it more probable that space is finite, and composed of units that the Ancient Greeks would have called "atoms" but perhaps what we call Planck lengths or quantums of length.

That is from the perspective of infinity, not necessarily from his own perspective (: It seems possible to me that the long spans of time between could diminish the effect of the undesired non-birthday. But then any speculation about what it is like to live forever is quite suspect. Just now I was thinking that perhaps immortality would not be so bad if you could ensure that you don't remember the same thing twice, only to lose myself in the idea that if one experiences every physical event in a finite space, they also experience every psychological one.
Even though I understand it, the fact still is quite strange, and morbid, that eventually everyone would choose to die.

Yes we could get into some trouble, since I have spent a lot of time on forums discussing philosophy and you have a strong interest in paradoxes involving the infinite xD

Just an alert that the answer to #5724 is stated in the question

Infinity is fun, isn't it? I have always thought so haha!

Once again, I don't think that it is much of a problem. If it isn't agreed otherwise, he can in fact always reply "Not yet!", because there will never come a day when it is declared that he cannot take all of the non-birthday pages with the pattern he is obeying.
I'm also not sure that Achilles would mind much if he was ordered to specify his page-tearing routine. He would just have to come up with a way of describing the largest possible finite number of days he can after which he is obligated to take a non-birthday page. And even if he only had a day to do so, he would be able to come up with a truly immense number. But then again, he would still have to suffer an infinite number of non-birthdays, so it's possible this is little compensation after all!

To be honest, I think that being immortal is the most frighteningly horrifying prospect imaginable, no matter what special powers you are given. To live until everything that could possibly happen in a finite space has happened, and then to do it again, and again, and again...

Zeno's spatial paradoxes once troubled me too, but eventually I got an explanation that satisfied me. Basically, very small distances are traversed in very small amounts of time, so time makes velocity possible. Zeno probably thought of time and motion as the same.

Yet again there are various solutions for this type of problem, yet clearly we are only expected to give the obvious one. That one is

Other solutions are (by substituting every combination of solutions): 1/5 and +/-4/5.