The worlds first riddle!

Mark:
Way off!!! The answer is approximately 2.5*10^158
Look at the first few terms:
100 + 100*99 + 100*99*98
These total 980,200, they're only 3% of the terms, AND they're growing. The last term is 100!

Yes, you are correct. As usual.
I must read the question more carefully in future.

A swimmer jumps from a bridge over a canal and swims 1 kilometer stream up. After that first kilometer, he passes a floating cork. He continues swimming for half an hour and then turns around and swims back to the bridge. The swimmer and the cork arrive at the bridge at the same time. The swimmer has been swimming with constant speed.

The Question: How fast does the water in the canal flow

CANAL
1 km/hr

PALINDROMIAL

What is p(x)?


Every solution is of the form a(x + 1)n, for a any constant, and n any positive integer.
Let p(x) = an xn+ an -1 xn - 1+ ... + a2 x2+ a1 x + a0 be a palindromial whose derivative p'(x) = nan xn - 1+ (n - 1)an - 1 xn - 2+ ... + 2a2 x + a1 is also a palindromial. From p(x) we get an= a0 and from p'(x) we get nan= a1.
Now an -1 is determined by p(x) and the value of a1, from which in turn a2 is determined from p'(x). Continuing in this fashion, we see that all coefficients are determined by the value of an.

Since multiplying a palindromial by a constant doesn't change its citizenship as a palindromial, with the same true of its derivative, we conclude that there is a unique palindromial with leading coefficient 1 whose derivative is also a palindromial. A moment's thought will verify that the polynomials (x + 1)n fit the requirements.


Mark:
CANAL
1 km/hr

It is obvious that the cork does not move relatively to the water (i.e. has the same speed as the water). So if the swimmer is swimming away from the cork for half an hour (up stream), it will take him another half hour to swim back to the cork again. Because the swimmer is swimming with constant speed (constant relatively to the speed of the water!) you can look at it as if the water in the river doesn't move, the cork doesn't move, and the swimmer swims a certain time away from the cork and then back. So in that one hour time, the cork has floated from 1 kilometer up stream to the bridge.

Conclusion: The water in the canal flows at a speed of 1 km/h.



Suppose a game of volleyball is tied at a score of 19 / 19. What is the probability that the game ends with a score of 21 / 19

Some particulars:
To win a game of volleyball a team must score 21 points and win by a margin of at least 2 points.

A team winning a rally either (a) scores a point if it served the ball, or (b) gains the right to serve the next rally if it received the ball.

In order to win a point a team must be serving the ball.




What is the smallest positive integer x so that 2168 divides x2003 + 1

PALINDROMIAL
Try wrote: "A moment's thought will verify that the polynomials (x + 1)n fit the requirements."

I'm not sure what this means. Do you agree with my answer that the coefficients of p(x) are found on the Nth line (where N is the order of p(x)) of Pascal's triangle?

In other words, if N is the order of p(x), then the coefficient of x^n is C(N, n) (N choose n).

VOLLEYBALL
Assuming each team has an even chance of winning each rally, the probability is 2/3.

One team (we don't care which or when) will go ahead 20-19 and will be serving. At that point, the probability of them (eventually) scoring again before the other team scores is 2/3.

2168 DIVIDES 2003X + 1

565

2168 DIVIDES 2003X + 1
Oh! Was that supposed to be x^2003 + 1? It looked like multiplication.

BOTTLE
Push in the cork.

ANT
First thought: 1/4, but I need to work on this.

ANT
Yep - Close enough to 1/4 to be indistinguishable by most computers.

True. Assuming your username is case-sensitive.

JUMPING BEETLES
I can get as low as 3.

TRACK TEAMS
I get 0.419 or 41.9%

DrewDad:
"True. Assuming your username is case-sensitive."

Clever


Mark:
JUMPING BEETLES
I can get as low as 3.

Color a 9x9 checkerboard so that the diagonals share the same color. Notice that since movement is diagonal, a beetle must jump to a square of the same color, thus allowing you to split the problem into smaller problems: the red beetles and the black beetles.
The red beetles can all be paired up, leaving no red squares with more than one beetle on it.

Now look at the black squares. Five rows of five squares alternate with four rows of four squares. Therefore, 25 beetles must be crammed into 16 spaces. Since at most 4 beetles can meet at a single square, the minimum number of squares where a beetle-meeting occurs is achieved when three squares have four beetles on them, and the rest have contain a single beetle, accounting for the total of 3•4 + 13 = 25 beetles.



TRACK TEAMS
I get 0.419 or 41.9%

Ditto





In how many ways can you change a
$50 bill using $20 bills, $10 bills,
$5 bills, $2 bills, and $1 bills




A man spent (in this order) a third of his life to date in the U.S., a sixth of it in India, 12 years in Egypt, half the remainder of his time in Australia, and as long in Canada as he spent in India.

Where did he spend his fortieth birthday




Dave drove at a steady clip along the highway, his wife beside him. "Have you noticed," he said, "that those annoying signs for Wild and Wonderful West Virginia seem to be regularly spaced along the road?
I wonder how far apart they are."

Sally glanced at her watch, then counted the number of signs they passed in one minute.

"What an odd coincidence!" exclaimed Sally "When you multiply that number by ten, it exactly equals the speed of your car in miles per hour."

Assuming that the car's speed is constant, that the signs are equally spaced, and that Sally's minute began and ended with the car midway between two signs, how far is it between one sign and the next

If I were him, I would like to spend it in a thebaid :wink:

40TH BIRTHDAY
In Egypt - he's 72

SIGNS
1/6 mile

Francis wrote, "If I were him, I would like to spend it in a the baid"

That reminds me of: A chicken and an egg are lying in bed. The chicken is smoking a cigarette with a satisfied smile on its face and the egg is frowning and looking a bit pissed off. The egg mutters, to no-one in particular, "Well, I guess we answered THAT question!"


40TH BIRTHDAY
In Egypt - he's 72

SIGNS
1/6 mile

Answer: The curious thing about this problem is that you do not need to know the car's speed to determine the spacing of the signs.

Let x be the number of signs passed in one minute. In an hour, the car will pass 60x signs. The speed of the car is 10x miles per hour. In 10x miles, it will pass 60x/10x, or 6 signs. The signs therefore are 1/6 mile apart.




This is guaranteed to drive you

Counting any way, except diagonally. How many times can you make the word: VALLEY



……........VALL
………....VALLE
……....VALLEY
……..VALLEY
…..VALLEY
..VALLEY
VALLEY

I'm not sure you got the meaning of "thebaid"

Thebaid. The last two comprised the upper part of the valley. During the fourth to fifth centuries it was the chosen land of the monks, who by their sanctity and by the form they impressed on the monastic system greatly influenced the East and the West.

Their monasteries may be divided into two groups. The best known is the Pachomian group, founded and legislated for by St. Pachomius. They formed a real religious order with Tabenna as a mother-house and its superior as their general. Besides Tabenna there were Peboou, Schenesit, Akhmin, Esneh, Monchosis, Thebaid, Tesmine, Hermopolis, and Armoutim. Saint Pachomius governed this group till his death (346), and was succeeded by Abbot Orcisius, and then by Abbot Francis.

I should have known you can walk on water.

Between you and me let's not have an "Adrian" wall