Mark wrote. “Note that if the problem is changed such that there are three possible outcomes (0, 1, 2) with probability 1/3 each, then both roots of the resulting polynomial are 1.”
That is so funny; change the question to fit the answer! Ha ha.
However I do agree with your reasoning and I don’t get that they die out….
It might be useful to observe that the rate of growth is expected to be 1.5 : in fact, the expected contribution to the population of each bacterium every minute is
−1⋅1/4 +0⋅1/4 +1⋅1/4 +2⋅1/4 = +0.5
Conditioning on the fact that the population has not extinguished up to time n , the expected number b n of bacteria at time n (starting from time 0 ) is
E( b n ∣ b 1 ,…,b n−1 ≠0 ) = (1.5) ^n
OR:
The probability of eventual extinction of a branching process is the smallest root in [0,1] of ϕ(t)=t , where ϕ is the probability generating function. In your case, ϕ(t)=(1+t+t^ 2 +t^ 3 )/4 , and the probability is 2 √ −1=.41421 .
Take your pick.
Something much easier; can you completely solve the following system of equations:
….........…....x + y + z + w = 10
.x^2 + y^2 + z^2 + w^2 = 30
x^3 + y^3 + z^3 + w^3 = 100
.......…………………..xyzw = 24
@Tryagain,
Gotta change the combination to my luggage from something other than 1234.
Perhaps to 4321.
Rap
By inspection (1,2,3,4) is a solution of the first and fourth equations and satisfies the second and third equations. Since the equations are symmetrical in x,y,z,w, the other 23 permutations of 1,2,3,4 are solutions also. But these are all the solutions, since the product of the degrees of the equations is 4!
So both Mark and Rap^ are correct, and receive a free milk shake with every purchase of a main meal in the A2K mess hall*.
*which as everybody knows is: A place on Federation and Klingon star ships or star bases where one can relax and enjoy a meal or a quick snack in a social setting… Enjoy!
Due to you guys always coming up with the right answer - I've been seeing a weather girl, thought it'd make a change to talk to someone that wasn't right all the time. Ha ha!
That was until she started to get to the root of the problem……before she will make out with me she wants to know the answer to…
How many negative roots does this equation have?
X^4 – 5x^3 – 4x^2 – 7x + 4 = 0
I would really appreciate some help here getting to first base!
@Tryagain,
According to Wolfram Alpha:
none.
Welcome to A2K Wolfram, it is good to see you giving Mark some excellent counsel.
The equation X^4 – 5x^3 – 4x^2 – 7x + 4 = 0 may be written in the form:(x^2 -2 )^2 = 5x^3 + 7x.
For every negative x, the left-hand member is non-negative and the right-hand member is negative, so NO negative x can satisfy the original equation.
As any of you guys with A2K security clearance will know I’m planning on entering the spring Go-Cart race down Pennsylvania Avenue.
However, today I dropped two wheel rims; one with a radius 15 intersects another with a radius 20 at right angles.
What is the difference of the area of the non-overlapping portions?
What can I say! The guy is a friggin’ genius. I always say, ‘If it’s worth doin’ at all, it’s worth taking the easy way!’
We can say in general, if any two areas a and b have a common area x, then the non-overlapping portions are a – x and b – x.
The difference of the areas of the non-overlapping portions is |a – b|.
In the problem poised, the difference is pi (20)^2 – pi (15)^2 = 175pi.
It is time once again for the prestigious A2K Squash Tournament.
This year there are n players in an elimination-type singles tournament.
How many matches will be played (or defaulted) to determine the winner?
Which by the grace of God and steroids will be me!
Accuracy and brevity; Mark has all three!
Each letter in the puzzle: A H H A A H / J O K E = H A
Uniquely represents a digit in the decimal scale.
What is the arithmetic division?
That is no joke Mark, that is indeed the correct answer.
I have it on good authority that Easter is almost upon us, and you may know there is a rumor that an A2K Moderator was seen entering a candy store where she bought ‘x’ Easter Eggs for ‘y‘ dollars’
( x and y are integers).
When she was about to leave I heard the clerk say, “If you buy ten more Easter Eggs I will give you all for $2 and you will save 80 cents a dozen.”
This sounds like a generous offer; but what are x and y ?
@Tryagain,
eggs:
There are at least two possible answers:
5 eggs for $1
50 eggs for $5
hidden /\
I have been called draconian and lugubriously Republican, but they are only some of my attributes. So Ima gonna say, ‘well done Mark’.
There is more than one way to view this problem and this is how I saw it…
Since y is an integer < 2, y = 1.
Then dealing with the price per egg in cents:
100 - 200 = 80
x....x+10 ...12
or:
x^2 + 25x – 150 = 0
Of which the positive root is x = 5, the number of eggs originally purchased.
Today I was trying to arrange a bilateral agreement when I thought that if an equilateral triangle and a regular hexagon have equal perimeters.
What is the ratio of their areas?
@Tryagain,
triangle and hexagon:
This can be done geometrically with no computation (other than counting small triangles):
Code:
/\
/ \
/____\
/\ /\
/ \ / \
/____\/____\
______
/\ /\
/ \ / \
/____\/____\
\ /\ /
\ / \ /
\/____\/
Well, how’s about that then!
DD gives the answer and Mark supplies the reason in pictorial glory - so even I can understand.
As you may know, I’m responsible for the A2K Cheerleaders dance routine and baton practice.
To ensure the girls don’t drop the baton, I propose that ten turns of a wire are helically wrapped around the cylindrical tube (with the outside circumference of four inches and a length of nine.)
The ends of the wire coincide with the ends of the same cylindrical element.
Is there anyone who cares enough to tell me how much wire I will need to avoid sweeping revisions to cheerleading safety rules?
BTW: I’m not Joe King; On January 27, 2009, in a lawsuit involving an accidental injury sustained during a cheerleading practice, the Wisconsin Supreme Court ruled that cheerleading is a full-contact sport in that state.
Of the United States' 2.9 million female high school athletes, only 3% are cheerleaders, yet cheerleading accounts for 65% of all catastrophic injuries in girls' high school athletics.
So three cheers for the cheerleaders, who are always willing to take one for the team!
Yet again Mark shows that Math can answer any question…
Roll the cylindrical surface and wire onto a plane. (Not a 747)
The element (9 inches), the repeated circumference (10.4 inches), and the wire now form a right triangle.
Hence, L=(81 + 1600)^1/2 or 41 inches of wire.
It has come to my attention that Mark is able to answer these questions over coffee or on the drive home!
To rectify this state of affairs, today’s enigma is taken from court transcripts:
The defendant (a renegade A2K member) had absconded with the office tea and bagel fund and decided to bury the tin on an island near the shore of which were two similar rocks A and B and farther inland, three coconut trees C1 C2 C3.
Stationing himself at C1 the defendant laid off C1 A1 perpendicular and equal to C1 A and directed outwardly from the perimeter of triangle A C1 B.
He similarly laid off C1 B1 perpendicular and equal to C1 B and also directed outwardly from the perimeter of triangle A C1 B.
He then located P1 the intersection of A B1 and B A1.
Stationing himself at C2 and C3, he similarly located points P2 and P3 and finally buried the tin at the circumcenter of triangle P1 P2 P3.
Returning to the island some years later the unrepentant bounder found that a big storm had destroyed all the coconut trees on the island!
Have you any ideas how I might find my swag?
