Marvin is playing a solitaire game with marbles. There are n bowls (for some positive integer n), and initially each bowl contains one marble. Each turn, Marvin may eitherremove a marble from a bowl, orchoose a bowl A with at least one marble and a different bowl B with at least as many marbles as bowl A, and move one marble from bowl A to bowl B.The game ends when there are no marbles left, but Marvin wants to make it last as long as possible.Prove that the game must end after at most n(n+1)/2 turns.Prove that for every positive integer k, there is an n such that Marvin can make the n-bowl game last for at least kn turns.Marisa comes along and asks to join the game. Marvin and Marisa revise the rules: they will alternate taking turns, starting with Marisa. When the game ends, whoever took the last turn is the winner.Prove that among any three consecutive values of n, there is at least one value for which Marvin has a winning strategy.
Answers
There is always at least one consecutive value of n for which Marvin has a successful approach.
By induction, we can demonstrate that the game must conclude after a maximum of n(n+1)/2 turns for the first half of the issue, Here Induction Means a method for demonstrating that a statement P(n) is true for every natural number n, and that the infinitely numerous examples P(0), P(1), P(2), P(3), etc. all hold, is mathematical induction. Informal analogies, such a ladder or a falling domino, are used to explain this technique .
Base case: If there is just one bowl, Marvin can get rid of it in one turn, ending the game after one.
Step one of induction: Assume that there are k bowls and that the game terminates after no more than k(k+1)/2 turns. When there are k+1 bowls, we shall demonstrate that the game terminates after at most (k+1)(k+2)/2 turns. The number of bowls lowers by one if Marvin takes a marble out of one, and the game must terminate after no more than k(k+1)/2 turns through induction hypothesis
When Marvin transfers a marble from bowl A to bowl B, where bowl A contains at least one marble and bowl B contains an equal number of marbles, bowl A loses one marble and bowl B gains one marble. The number of bowls stays the same because bowl B has at least as many marbles as bowl A. As a result, the game must still terminate after k(k+1)/2 turns at the latest.
In any scenario, the game ends after at most k(k+1)/2 turns, thus when there are k+1 bowls, the game must end after at most (k+1)(k+2)/2 turns. For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.
Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.
For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.
Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.
Marvin can select a bowl containing at least k marbles on each turn and transfer one marble to a bowl containing fewer than k marbles .Marvin can transfer one marble from the first bowl to the second bowl, for instance, if there are k+1 marbles in the first bowl and k marbles in the second bowl. This guarantees that there are at least k marbles in each bowl. The game will go for at least kn turns because Marvin can take at least one turn every bowl.
For the third component of the issue, we can demonstrate that there is always at least one value of n for which Marvin has a successful strategy.
Assume that n is a positive integer. We shall demonstrate that either Marvin or Marisa has a winning strategy for the game of n bowls or the game of (n+1) bowls.
If Marvin has a winning strategy for the N-bowl game, he can use that method to win the N-bowl game.
If Marisa has a winning strategy for the n-bowl game and Marvin does not, then Marisa does as well. In this scenario, Marvin can use Marisa's successful approach to the n-bowl game as his own successful approach to the (n+1)-bowl game.
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Related Questions
What is equivalent expression to 2a
Answers
Answer:
2a
Step-by-step explanation:
up at the top is the answer
Help me with this please
Answers
The two pairs of Adjacent angles in the figure are:
∠1 and ∠3, ∠3 and ∠7.
The correct option is c.
In the given diagram with two intersecting lines and angles ∠1, ∠3, ∠5, and ∠7, the adjacent angle pairs are as follows:
Adjacent angles to ∠1:
∠1 and ∠5
∠1 and ∠3
Adjacent angles to ∠3:
∠3 and ∠1
∠3 and ∠7
Adjacent angles to ∠5:
∠5 and ∠1
∠5 and ∠7
Adjacent angles to ∠7:
∠7 and ∠3
∠7 and ∠5
Adjacent angles are angles that share a common side and a common vertex, where the vertex is the point of intersection between the two lines. In this case, the adjacent angle pairs can be identified based on their relationship to each angle (∠1, ∠3, ∠5, ∠7) and the intersecting lines.
from the given choices,
∠1 and ∠3, ∠3 and ∠7
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Find product of face values of all digits in
the number 91250
Answers
To find the product of the face values (the values of the digits as they appear in the number) of all digits in 91250, we can simply multiply the face values of each digit together.
The face value of a digit is simply the value of the digit itself, regardless of its position in the number.
So, we have:
9 * 1 * 2 * 5 * 0 = 0
Note that the face value of the digit 0 is 0, so any product that includes a 0 will also be 0.
Therefore, the product of the face values of all digits in 91250 is 0.
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What is the exponent for the expression 6 × 6 × 6 × 6 × 6?
Answers
Answer:
5
Step-by-step explanation:
6*6*6*6*6=\(6^{5}\)
Answer:
\(\displaystyle 5\)
Step-by-step explanation:
The quantity 6 is being multiplied five times, therefore you have your answer. That is to say, the exponent tells the base [6] to multiply itself as many times as intended [5].
I am joyous to assist you at any time.
4/3 divided by 12= simplified and fraction form!
Answers
Answer:
\({\huge\pink{\fbox{{࿐αɴѕωєя࿐}}}}\)
\( \frac{4}{3} \div 12 \\ = \frac{4}{3} \times \frac{1}{12} \\ = \frac{4}{3 \times 12} \\ = \frac{4}{36} \\ = \frac{1}{9 } \\ = 0.11\)
ʰᵒᵖᵉ ⁱᵗ ʰᵉˡᵖˢ
\( \huge\red{ \mid{ \underline{ \overline{ \tt ꧁❣ ʀᴀɪɴʙᴏᴡˢᵃˡᵗ2²2² ࿐ }} \mid}}\)
Answer:
1/9 is the answer
4/3 x 1/12 just multiply but the reciprocal
hope this helps
have a good day :)
Step-by-step explanation:
The best bagel company bakes 1,872.they put a bakers dozen ,or 13 bagels, into each bag. How many bags do they fill
Answers
Answer:
144
Step-by-step explanation:
1872 divided into bags with 13 each is
1872/13= 144 bags to fill
I need help with #21
Answers
Answer: x=9
Step-by-step explanation:
(5x+7)
5 x 9 = 45
45 + 7 = 52
Find the general indefinite integral. (Use C for the constant of integration.) (x^1.4 + 7x^2.5) dx
Answers
The general indefinite integral is: (x^2.4)÷2.4 + 2 × x^3.5 + C.
How we find the general indefinite integral of the given function?To find the indefinite integral of (x^1.4 + 7x^2.5) dx, you'll need to perform integration on each term separately and add a constant of integration (C).
For x^n, the integral is (x^(n+1))/(n+1).
Integrate x^1.4.
The integral of x^1.4 is (x^(1.4+1))/(1.4+1) = (x^2.4)/2.4.
Integrate 7x^2.5.
The integral of 7x^2.5 is 7 × (x^(2.5+1))/(2.5+1) = 7 × (x^3.5)/3.5 = 2 * x^3.5.
Combine the integrated terms and add the constant of integration (C).
The general indefinite integral is: (x^2.4)/2.4 + 2 × x^3.5 + C.
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Could i get some help with these math questions?
Answers
69 ayyy lol
What is the solution to the equation |y/4-2| = 22
Y=-6
Y=-5
Y=80
Y=96
Answers
I hope this helps. If you can, please give me brainliest.
Solve the augmented matrix by elementary row operations. 9. (4 points) Let A and B be 3 by 3 matrices with det (A) = 3 and det (b) = 5. Find the value of det (AB).
Answers
The value of determinant of the matrix det (AB) is 15.
Given matrices A and B are 3 by 3 matrices with
det (A) = 3 and
det (b) = 5.
We need to find the value of det (AB).
Writing the given matrices into the augmented matrix form gives [A | I] and [B | I] respectively.
By multiplying A and B, we get AB. Similarly, by multiplying I and I, we get I. We can then write AB into an augmented matrix form as [AB | I].
Therefore, we can solve the augmented matrix [AB | I] by row reducing [A | I] and [B | I] simultaneously using elementary row operations as shown below.
The determinant of AB can be calculated as det(AB) = det(A) × det(B)
= 3 × 5
= 15.
Conclusion: The value of det (AB) is 15.
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We need to find the value of determinant det(AB), using the formula: det(AB) = det(A)det(B)
=> det(AB) = 3 × 5
=> det(AB) = 15.
Hence, the value of det(AB) is 15.
The given matrices are A and B. Here, we need to determine the value of det(AB). To calculate the determinant of the product of two matrices, we can follow this rule:
det(AB) = det(A)det(B).
Given that: det(A) = 3
det(B) = 5
Now, let C = AB be the matrix product. Then,
det(C) = det(AB).
To evaluate det(C), we have to compute C first. We can use the following method to solve the augmented matrix by elementary row operations.
Given matrices A and B are: Matrix A and B:
[A|B] = [3 0 0|1 0 1] [0 3 0|0 1 1] [0 0 3|1 1 0][A|B]
= [3 0 0|1 0 1] [0 3 0|0 1 1] [0 0 3|1 1 0].
We can see that the coefficient matrix is an identity matrix. So, we can directly evaluate the determinant of A to be 3.
det(A) = 3.
Therefore, det(AB) = det(A)det(B)
= 3 × 5
= 15.
Conclusion: Therefore, the value of det(AB) is 15.
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NEED HELP ASAP ASAP!!!!!
Answers
In which interval is the median?
A sample of 50 students was surveyed about their time
spent in front of a screen (laptop, tablet, phone, etc.).
Each interval contains the left endpoint but not the
right endpoint.
[4,5)
[5,6)
[6,7)
[7,8)
Hours of Screen Time per Day
10
B
Frequency
2
6
8
10 12
Answers
Answer:
B is correct! Took the quiz 2021 <3. Yall b safe!
Step-by-step explanation:
what data do scientists use to determine when a volcano might erupt?
Answers
What is the slope of the graph below?
Answers
Answer:
-3/2
Step-by-step explanation:
each day you read 15 pages in your textbook. use the drop-down menu to choose the correct equation that represents the total number of pages you have read after any number of days.
Answers
Answer:
15x
Step-by-step explanation:
assum 1 day you read 15 pages then after X day you reads 15*X=15X
Please Solve below A. Express the vector in the form v = v₁i + v₂j + v3k. 8u-5 vif u = (1, 1, 0) and v= = (3, 0, 1) Ov=23i+8j - 5k v=-7i+13j - 5k Ov=-7i+8j - 5k v=8i + 8j - 5k B. Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2) and R(2, 2, 1). O √64- -(i-2j+ k) 06(1-21-k) O (i-2j-2k) 06(1-2j-k)
Answers
A. The vector v expressed in the form v = v₁i + v₂j + v₃k is v = -7i + 8j - 5k. B. A unit vector perpendicular to the plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2), and R(2, 2, 1) is (i - 2j - 2k).
A.To express the vector v in the form v = v₁i + v₂j + v₃k, we simply substitute the given values of v₁, v₂, and v₃. From the given options, the vector v = -7i + 8j - 5k matches the form v = v₁i + v₂j + v₃k.
B. A unit vector perpendicular to the plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2), and R(2, 2, 1) is (i - 2j - 2k).
To find a unit vector perpendicular to a plane, we need to find the cross product of two vectors that lie in the plane. We can find two vectors in the plane PQR by taking the differences between the coordinates of the points: PQ = Q - P = (1 - 2)i + (1 - 1)j + (2 - 3)k = -i - k, and PR = R - P = (2 - 2)i + (2 - 1)j + (1 - 3)k = j - 2k.
Taking the cross product of PQ and PR gives us the vector (-1)(1)k - (-1)(-2)j + (-1)(-1)(-i) = -k + 2j - i. To make this a unit vector, we divide it by its magnitude. The magnitude of the vector is √((-1)² + 2² + (-1)²) = √6. Dividing the vector by √6, we get the unit vector (i - 2j - 2k). Therefore, a unit vector perpendicular to the plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2), and R(2, 2, 1) is (i - 2j - 2k).
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Given the relation R = {(-2, 3), (a, 4), (1,9), (0,7)}.
Which replacement for a makes this relation a function
O4
0-2
00
Answers
Answer:
The answer would be 4
Step-by-step explanation:
you can’t have repeating numbers in a function.
the distance from the ground of a person riding on a ferris wheel can be modeled by the equation d equals 30 times the sine of the quantity pi over 40 times t end quantity plus 20 comma where d represents the distance, in feet, of the person above the ground after t seconds. how long will it take for the ferris wheel to make one revolution? 30 seconds 40 seconds 80 seconds 20 seconds
Answers
The equation given models the distance from the ground of a person riding on a ferris wheel. it takes 80 seconds for the Ferris wheel to make one revolution.
To determine how long it will take for the ferris wheel to make one revolution, we need to find the period of the function. The period is the amount of time it takes for the function to complete one full cycle.
In this case, the function is d = 30sin(pi/40t) + 20, where t is measured in seconds. The period of the function can be found using the formula T = (2pi)/b, where b is the coefficient of t in the argument of the sine function. In this case, b = pi/40, so T = (2pi)/(pi/40) = 80 seconds.
Therefore, it will take 80 seconds for the ferris wheel to make one full revolution. The answer is option C, 80 seconds.
The time it takes for a Ferris wheel to make one revolution can be determined using the given equation: d = 30 * sin((π/40) * t) + 20. In this equation, d represents the distance (in feet) of the person above the ground, and t represents the time in seconds.
A full revolution occurs when the angle inside the sine function completes a cycle of 2π radians. To find the time it takes for this to happen, we need to equate the angle (π/40) * t to 2π:
(π/40) * t = 2π
To solve for t, we can divide both sides of the equation by (π/40):
t = 2π * (40/π)
The π in both the numerator and denominator cancels out:
t = 2 * 40
t = 80 seconds
Therefore, it takes 80 seconds for the Ferris wheel to make one revolution.
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A farmer has enough feed to last his 15pigs for 20days. How long would the feed last if he had 10pigs
Answers
40/3 days, (13.33 days)
Step-by-step explanation:
Set up the equation, 15/20 = 10/x, and then use cross mutiply to solve it.
10 * 20 = 15x, x = 200/15 = 40/3
Why?
b/c if each pig eats the same amount of food, then the ratio between the number of pigs and days should remain constant.
3 points
15. Metal A has a density of 2.4g/cm3 and Metal B has a density of 1.8
g/cm3. Their masses are combined in the ratio 2:3 to make metal C which
has a mass of 60 grams. Find the density of the metal C*
Answers
Answer:
The density of the metal C is 2 g/cm³
Step-by-step explanation:
The parameters of the two metals are;
The density of the Metal A = 2.4 g/cm³
The density of the Metal B = 1.8 g/cm³
The ratio in which their masses are combined to make the metal C= 2:3
The mass of the metal C, m = 60 grams
Let, 'x', represent the mass of the metal A in the 60 grams of the metal C and let 'y' represent the mass of the metal B in the 60 grams of the metal C
Let the ratio of the metals A and B in the metal C = x:y = 2:3
Therefore;
The mass of the metal A in the metal C, x = (2/(3 + 2)) × 60 g = 24 g
The volume of the metal A in the metal C, V₁ = (24 g)/(2.4 g/cm³) = 10 cm³
The mass of the metal B in the metal C, y = (3/(3 + 2)) × 60 g = 36 g
The volume of the metal B in the metal C, V₂ = (36 g)/(1.8 g/cm³) = 20 cm³
The volume of the metal C, V = V₁ + V₂ = 10 cm³ + 20 cm³ = 30 cm³
The density of the metal C = ρ = m/V
ρ = (60 g)/(30 cm³) = 2 g/cm³
The density of the metal C, ρ = 2 g/cm³
The baker is able to make a perfect pastry 97 of the time If he bakes 300 pies what is the probability that at least one doesn't turn out?
Answers
Answer:
there is a a 3% chance one doesnt turn out
Step-by-step explanation:
There is a one percent chance because 3(97/100)= 291/300 simplified we get back to 97/100 which is equal to 97%, 100-97=3 giving us 3%.
3y + 2 = 2y - 5x
y = mx + b form:
Answers
answer:
y= -5x -2
hope it helps
4688-3) -30X-5 CX-5)
Answers
Answer:
x = 13/3
Step-by-step explanation:
4(8x-3) - 30x = 5(x-5)
Distribute
32x - 12 - 30x = 5x -25
Combine like terms
2x -12 = 5x-25
Subtract 2x from each side
2x-12-2x = 5x-2x -25
-12 = 3x-25
Add 25 to each side
-12+25 = 3x-25+25
13 = 3x
Divide each side by 3
13/3 = x
Answer:
x ≈ 4.3
Step-by-step explanation:
Distribute the 4 to (8x - 3) and the 5 to (x - 5), like so:
4(8x - 3) - 30x = 5(x - 5)
32x - 12 - 30x = 5x - 25
Add like terms (32x and -30x), like so:
32x - 30x = 2x
2x - 12 = 5x - 25
Subtract 2x from both sides:
2x - 12 = 5x - 25
-2x -2x
____________
-12 = 3x - 25
Add 25 to both sides:
-12 = 3x - 25
+ 25 + 25
____________
13 = 3x
Divide both sides by 3:
4.3 ≈ x
Find An Equation Of The Plane That Contains All The Points That Are Equidistant From The Given Points. (-7, 3, 1), (6, -2, 4)
Answers
An equation of the plane that contains all the points that are equidistant from (-7, 3, 1) and (6, - 2, 3) is,
⇒ 13x - 11y + 12z = 105
Now, Let (x, y, z) be a point on the plane that contains all the points that are equidistant from (-7, 3, 1) and (6, -2, 4).
Then, the distance from (x, y, z) to (-7, 3, 1) is equal to the distance from (x, y, z) to (6, -2, 4).
Hence, By Using the distance formula, we get:
√[(x - (-7))² + (y - 3)² + (z - 1)²]
= √[(x - 6)² + (y + 2)² + (z - 4)²]
Squaring both sides, we get:
(x - (-7))² + (y - 3)² + (z - 1)²
= (x - 6)² + (y + 2)² + (z - 4)²
Expanding and simplifying, we get:
13x - 11y + 12z = 105
Therefore, an equation of the plane that contains all the points that are equidistant from (-7, 3, 1) and (6, - 2, 3) is,
⇒ 13x - 11y + 12z = 105
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Find the slope and the y-intercept of the graph of y + 1 = x.
Answers
Answer: The slope is 4/3
Step-by-step explanation:
Whatever number is attatched to the "x" is the slope.
Answer:
Slope: M = 4/3
Y-Intercept: : (0,−1)
Step-by-step explanation:
Find the exact length of the third side.
4
2
Answers
Answer:
Step-by-step explanation:
4-2<x<4+2
2<x<6
x = 3, 4 or 5
i assume u r talking about triangle
For what values of x is the expression below defined?
Sqrt x+5 divided by sqrt 1-x
Answers
Answer:
D. - 5 ≤ x < 1
Step-by-step explanation:
The numerator cannot be the square root of a neg. no.
So, x + 5 ≥ 0 or x ≥ -5
The denominator cannot be a neg. no. or 0.
So, 1 - x > 0
1 > x or x < 1
So, altogether - 5 ≤ x < 1
Therefore, D is the correct answer.
what is this fraction in its simplest form?
Answers
Answer:
a^2/3
Step-by-step explanation:
the numerator simplifies to 5a^3 and the bottom is 15a. we can use our exponent rules to subtract exponents, so it is 1/3(a^2). this simplifies to a^2/3.
Answer:
a^2
------
3
Step-by-step explanation:
5 a^3 / 15a
Simplify the numbers
5/15 = 1/3
Simplify the variables
a^3 /a = a^2
Putting it back together
a^2
------
3
Solve problem in photo 8th math
Answers
Answer:
I did this test it Is b
Step-by-step explanation:
Answer:
The answer is B 16
Step-by-step explanation:
The square root of it is 15.9059737206 but if you round it the closest answe is B.