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

Answer 1

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.

To learn more about induction at: brainly.com/question/8419798

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Related Questions

PLS HELP!!!
Find x M 13 P х 18° R .02 40 42.07 4.22​

Answers

just add them up EZZZZZZZZZZZ

List three integers that are between -22 and -10
PLEASE HELP ASAP

Answers

Answer:

-21, -20, -19, --18, -17, -16, -15, -14, -13, -12, -11

Step-by-step explanation:

There is no step by step explanation.

The temperature of a hot steak starts at 140f and decreases at 5f per minute what will the temperature of this steak after a certain number of minutes

Answers

Answer:

T(m) = (140 - 5m)

Step-by-step explanation:

Given that :

Initial temperature, a = 140 F

Rate of decrease = 5 F per minute

Given a certain Number of minutes, m

Temperature, T after m minutes ;

Using the formula :

Final temperature = Initial temperature - 5*number of minutes)

T(m) = (140 - 5m)

Given that ; m = 10

T(10) = 140 - 5(10)

T(10) = 140 - 50

T(10) = 90 F

Find the area of the semicircle

Answers

Answer:

25.13 m

Step-by-step explanation:

pi r²÷2

pi 4²÷2=25.13

3 − 1 = 5 + 10 = −5 1/2 Check whether the given value of is a solution to the equation. Justify your answer.

Answers

Answer:

[tex]x = -5\frac{1}{2}[/tex] is the solution

Step-by-step explanation:

Given

[tex]3x - 1 = 5x + 10[/tex] --- Correct expression

[tex]x = -5\frac{1}{2}[/tex]

Required

Determine if the given solution is correct

To do this, we simply solve for x in the given equation.

So, we have:

[tex]3x - 1 = 5x + 10[/tex]

Collect like terms

[tex]3x - 5x = 1 + 10[/tex]

[tex]-2x = 11[/tex]

Divide both sides by -2

[tex]\frac{-2x}{-2} = \frac{11}{-2}[/tex]

[tex]x = -\frac{11}{2}[/tex]

[tex]x = -5\frac{1}{2}[/tex]

which of the following is a exponential function ?

Answers

The third one is exponential function

answers r
268
201
151
67

Answers

67in³

Answer:

we have

volume of sphere =4/3 πr³=1/3×π×4³=67.02=67in³ is your answer.

Find the perimeter of this complex figure

Answers

Answer:

38

Step-by-step explanation:

9 + 10 + 2 + 6 + (9 - 2) + (10 - 6) = 38

Is 9.90 greater than or lesser than 9.9

Answers

Answer:

it is equal too.

Step-by-step explanation:

Answer:

it is equal to

Step-by-step explanation:

Which equation does this model represent

Answers

Answer:

G.

Step-by-step explanation:

5 + (-9) = -4.

. .

. .....

Newton's law of gravitation state that the force F of attraction between two bodies varies jointly as their masses M1 and M2 and inversely as the square of the distance between them. Two bodies whose centers are 500mi apart attract each other with a force of 15lb.What will be the force of attraction if their masses were tripled and the distance between their centers doubled?​

Answers

I really don’t know this answer I tried everything

In triangle ABC, B = 120°, a = 10, c = 18
Find angle C

choices
A. 40°
B. 50°
C. 45°
D. 20°

Answers

Answer:

kk

Step-by-step explanation:

Please help no link as answers

Answers

Step-by-step explanation:

You can imagine this figure as a rectangle and cube

If you want volume of this irregular figure than you have to do it like this:

V(figure)= V(rectangle)+ V(cube)

V(figure)= a*b*c+ a³

V(figure)= 4*3*(I don't see dimension on the left)+ 3³

V(figure)=12*(I don't see dimension on the left)+ 27

And only you have to to do is to set this dimension which I can't see.

NO LINKS FOR THE LAST TIME JUST HELP ANSWER THE QUESTION WITH A B C OR D WILL GIVE BRAINLIEST!!

Answers

Answer: letter b

Step-by-step explanation:

Which of the following circumstances describes a proportional relationship?
Water flows from a hose into an empty 10 gallon bucket at a rate of 2 gallons per minute.
plumber is paid $50 per hour plus a $75 service charge.

The cost of renting a car is $45 for the first day and $20 per day for each additional day.

A student has $100 in a savings account and saves $5 per month for the next twelve months.

Answers

bStep-by-step explanation:

Find an expression in its simplest form for the perimeter of this shape

Answers

Answer:

(18p + 18)

Explanation:

the perimeter of the shape is the sum of all its sides. we are given the individual lengths of all the sides except for the horizontal one at the bottom and the vertical one between (2p + 5) and (3p + 4), but they can easily be calculated using the other values.

the horizontal length at the bottom is the sum of (2p + 5) and (3p + 4), or (5p + 9).

the vertical length is the difference between 4p and 2p, which is 2p.

now, we can add all of these values together:

4p + (2p + 5) + 2p + (3p + 4) + 2p + (5p + 9)

= 4p + 2p + 2p + 3p + 2p + 5p + 5 + 4 + 9

= 18p + 18

i hope this helps! :D

What is the measure of ABC?

Answers

I am not absolutely sure but it should be 310

mAC equals to 50*2=100°
So the measure of ABC is:
360-100=260° (B)
I Hope this helped! Good luck ^^

Three friends were helping to push a car. If all three are pushing equally hard and the net force is 600N, how hard is each one pushing the car?

Answers

Answer:

One person is pushing the car in the net force of 200N

Step-by-step explanation:

600 divided by 3 (600 N in total, and 3 person pushing.)

24 is what percent of 400?

Answers

Answer:

The answer is 6%

Answer:

6%

Step-by-step explanation:

tell me if I’m wrong lol

Solve for x.
please help real quick

Answers

Answer:

(7/2, 0)

Step-by-step explanation:

Answer:

x=11

Step-by-step explanation:

6x-21=45

6x=66

x=11

In which step did John first make a mistake?
look at photos

Answers

Answer:

John did not make a mistake

Step-by-step explanation:

he did everything correct

he didn’t make a mistake

find m the in this screenshot

Answers

Answer:

[tex]x + x + 4 + 3x - 9 =90 \\ 5x = 95 \\ x = 19 \\ \\ gjh = 19 + 4 = 23[/tex]

Please solve the question in the picture below.

Answers

Answer:

a

Step-by-step explanation:

because y=mx+b m =slope and b is the y intercept

Answer:

A. is the right answer

slope is 1.5 and y intercept is -2

pls pls pls help me giving brainlist

Answers

Answer:

The first one is a similie

second is a metaphor

Step-by-step explanation:

like ice means like a hatred glare

the hug is a a good thing thats

happens on a bad dat

Answer:

simile

the referee was angry

metaphor

they have a lot of love for each other

Step-by-step explanation:

please mark me branliest

Two students spent the same amount of
money on their lunches and got the same
number of toppings. One bought a steak
burger and paid $3.80 plus $0.40 for each
topping. The other bought a hamburger
and paid $3.20 plus $0.60 for each
topping. How many toppings did each
student get?

Answers

0.60-0.40=0.20

3.80-3.20=0.60

0.60 divided by 0.20=3

Hence, each student got 3 toppings each.

hope it helps:))

The company Apple is worth approximately 8×10^9 dollars. The company Jackson Avenue Coffee is worth approximately 2×10^5 dollars. How many times bigger is Apple's worth than Jackson Coffee's worth?​

Answers

Answer:

35

Step-by-step explanation:

Jayden went to the shop and bought and orange juice for $150 and a bun for $90 . He told the shopkeeper he didnt want any coins below $10.00 nor did he want any one hundred dollar notes.

Jaydens Change included?​

Answers

Answer:

give 300 and get back 60?

Step-by-step explanation:

Draw a line through the point (2, 2) with a slope of 1. Draw a line through the point (1,−1) with a slope of 1.

Answers

Answer:

should look like this

Step-by-step explanation:

equations are y=x  and  y = -x if needed

The equation of the lines are y = x and y = x - 2

A linear equation is given by:

y = mx + b;

where y, x are variables, m is the slope of the line and b is the y intercept

The equation of a line passing through the point (2, 2) with a slope of 1 is:

[tex]y-y_1=m(x-x_1)\\\\y-2=1(x-2)\\\\y = x[/tex]

The equation of a line passing through the point (1, -1) with a slope of 1 is:

[tex]y-y_1=m(x-x_1)\\\\y-(-1)=1(x-1)\\\\y = x-2[/tex]

Therefore both lines have equation of y = x and y = x - 2. The graphs of the lines are attached.

Find out more at: https://brainly.com/question/1391192

How to solve systems of inequalities

Answers

Answer:

Determine whether an ordered pair is a solution of a system of linear inequalities.

Solve a system of linear inequalities by graphing

Solve applications of systems of inequalities

You can use desmos

Step-by-step explanation:

Graph x>2 on a number line.

Solve the inequality 2a<5a+12.

Determine whether the ordered pair \left(3,\frac{1}{2}\right) is a solution to the system \left\{\begin{array}{c}x+2y=4\hfill \\ y=6x\hfill \end{array}.

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

System of Linear Inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.

\left\{\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs \left(x,y\right) that make both inequalities true.

Solutions of a System of Linear Inequalities

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the x-y coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system. s

Solve 8y - 9 = -3y + 2​

Answers

Answer:

y=1

Step-by-step explanation:

8y-9=-3y+2

-2

8y-11=-3y

-8y

-11=-11y/-11

Answer:

y=1

Step-by-step explanation:

8y+3y=2+9

11y/11=11/11

y=1

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