Prime

Is that because..

If I made finals, I would choose the line in which I already ran and won. Another possibility is to make a note, which is the preferred line of the favorite in the competition, and grab his line. Another more serious, variation -- to choose the line next to favorite (if he gets to choose after you), so that you can compete close to him. Also, isn't 200 m run in a straight line without any turns?

Correction: I meant B*t*ln(t+e) as the function for the distance traveled by Bug (starting at d=0 from the wall). And the instanteneous speed: B*ln(t+e) + B*t/(t+e). The puzzle here, to find justification for the instanteneous speed formula.

But the question says 'how long'.... There is every chance that my maths is wrong here somewhere so please feel freee to correct, but it feels as though its about right.... That's a good solid effort supplying Bug with necessary differential equations helping to estimate the time it would take to catch up with the scooter. It does seem that the time required is to the tune of e100. However, I have some problems with the solution. 1. I disagree with the conjecture that "The Prof needs to start a distance d from the wall initially..." just because "Otherwise, the bug has already caught up and my equations won't work as they are undefined for d=0 at t=0." It is a very viable everyday occurrence that Prof starts at 0 metres from the wall instantly attaining speed of 1m/sec and the magic rubber band and zero-dimention Bug spring into existence. If equations don't work for that situation, we must find different equations. 2. You found a function describing distance traveled by Bug: s = Bu*ln(Au/d), or expanding variable "u": s = B*(t+d/A)*ln(tA/d+1) Where B is Bug's speed; A is the speed of scooter; d is initial distance of scooter from the wall; and t is elapsed time. In our problem we set d=1 metre arbitrarily. But what if d was much smaller? Something like d=10-44? Then after 1 second (t=1), Bug would have traveled more than 1 metre -- further than the scooter! That disagrees with our notion of how long it takes Bug to catch up! It seems that function describing the distance traveled by Bug: s = B(t+1)*ln(t+e) fits the bill. But if we take the derivative s' = B*ln(t+e) + B(t+1)/(t+e) -- it does not agree with our concensus what instanteneous speed for Bug should be. (Even though I suggested instanteneous speed myself post#15). P.S. I have a feeling, there is a way to solve that problem without use of differential equations.

I guess "expected mean" means half the time it takes to carve the name counting from the time someone started to carve the name. In that case the killers are: Next time, please choose simpler names for your archeologists.

Are we allowed the use of double integral and natural logarithm for solving this puzzle? Do we allow puzzles that require the knowledge of advanced calculus on this forum? Without use of advanced calculus, I can give you only half solution for now. I'll try to find simple non-calculus way to solve the puzzle. But if you are not aware of such way yourself, please let us know. <_<

1. I can walk faster, than Prof. T. rides his scooer. 2. The initial length of the string is not mentioned, nor the length of the bug. If the bug can reach the rear fender from start, it can simply hang on to it.

That's a tough math.

Nice algebra problem.

Is that what you were going for? Your answer is essentially correct. Here is how I'd formulate it:

THAT'S A GOOD GUESS! Here are some examples:

Forget negative. Cosider all digits positive. I should have stipulated that in the OP. It's my omission. As a side note: If you cannot defeat it -- use it.©Prime.

When an infinite series does not converge, it does not mean that it's not defined. If it was not defined, we wouldn't be able to discuss it intelligently. Mathematicians have been tackling with understanding of infinity for centuries and came to a certain concensus in some respects and not so much in others. The OP gives an illustration of a typical mapping problem. With such problems the mathematical concensus has been reached long time ago. Regard famous Galileo's paradox, where two infinite sets, one of all natural numbers, anoher -- of all their squares are mapped one to one. 1 --> 1 2 --> 4 3 --> 9 4 --> 16 and so on. On one hand, there is one to one correspondence between the two sets, on the other hand, second set does not seem to have some numbers that the first set has. Which is bigger? The thing is, assigning numbers to the memebers of an infinite set, does not change the nature of that set. You may or may not be able to do so (assign numbers). Since some infinite sets are, indeed, larger than others. A typical example is an infinite set of all natural numbers versus infinite set of all real numbers between zero and 1. The latter is larger than the former. That is, you cannot make one to one correspondence between the set of real numbers (0<=r<1) and set of all natural numbers from 1 to infinity, like we did in Galileo's paradox above. Some real numbers will always be left unpaired with natural number. But tat's more advanced topic. With respect to the OP, we are dealing wih infinite sets of the same order here, where one to one correspondence can always be found. One property of infinite set, which mathematicians had agreed upon long time ago: you can divide infinity by any finite number and the result would be an infinity. Furthermore, the resulting infinite set is not "smaller" than the original set. I'll refer you to another well known example -- Hilbert's Hotel. Famous mathematician David Hilbert used it as an illustration in his lectures (a century or so ago). The variation that I like goes like this: In a small town N, there is an hotel with infinite number of rooms, all of which are occupied. There happens to be a superball game in that town, and an infinite number of fans come to watch the game. As a manager of that hotel, how would you place the newcomers? In conclusion, I'd recommend to reading "Infinity and the Mind" by Rudy Rucker to all who is interested in the subject. It's a popular, yet in-depth well-written book.

Here is a simple re-mapping procedure: 1. Every time you remove 2 balls from bin A, chose the next two numbers not divisible by 3. E.g., (1,2), (4,5), (7,8)... etc. 2. Upon placing removed balls into the bin B, wipe out the numbers and initialize the balls with new numbers sequentially. (1,2), (3,4), (5,6), ... etc. 3. Throw away odd numbered ball from each pair. Those operations are equivalent to the OP in terms of the number of balls moved and thrown away. After one minute has elapsed and you have performed an infinite number of operations you'll have an infinite number of even numbered balls in bin B; and infinite number of balls with numbers divisible by 3 plus some more in bin A.

Ben asked to find a number, which is a cube of the sum of its digits. Clue ran a program performing some exhaustive search under 1,000,000 and found that the cubes of 0, 1, 8, 17, 18, 26, and 27 have that property. I claim that I can find more numbers with that property even under 100,000. And I don't need to use a computer for that. Help my argument.

You could give me a benefit of a doubt. And treat my proposition as a puzzle. May be I'll make a separate post of it.

OK, so the the question #2 in my previous post may be too mathematical, so we don't want to tackle it here. But, surely, you can solve my question #3. That's just a puzzle.

Why alcohol is the driving force behind mathematical discovery at Morty's? Patrons remain largely indifferent to Alex's challenges, until he offers to buy drinks. Can we implement something similar at this forum?

1. You've missed 0 again. 2. Can there be any numbers greater than 1 million with that property? Better yet, any numbers greater than 157,464? 3. I can find many more numbers with that property, not shown in your "exhaustive list", all under 100,000. And I can find those without aid of a computer.(That's a new puzzle).

Is zero negative, or is it not an integer?

While it's a commercial break, here are couple more: Interesting question: what is the largest possible such number?

I found 5 such numbers. After which I gave up further search and went back to watching olympics.

What do you mean only 4? I found 6.

I see now that my statement of the problem is too confusing. The essence of the problem is that one man sets the odds, another man chooses which side he bets on. For example, CR says: "The light is blue, the odds for "payoff" event are 1:2". I say: "OK, then I bet my 1 rulbe against your 2 rubles that the machine will spew change next time someone pulls the lever." Or the light is pink and the odds according to CR are 3:4. Then I bet my 4 rubles against CR's 3 that the machine will NOT payoff next time someone pulls the lever.

Are you overlooking betting odds we set: 1:2 and 3:4?