Answer:
Given that there are 21 flags, NOW using backward induction we can see that;
If there is only one flag left, whosoever picks up that particular flag will lose. Therefore, 1 is loosing number.
Now if there are 2 flags left, whosoever's turn it is, can pick 1 flag and leave one for the opponent and win the game. Therefore, 2 is winning number. Similarly if there are 3 flags left then whosoever's turn it is can pick 2 flags and leave one for the opponent and can win the game. Hence 3 is winning number. Similarly when 4 flags left, 3 can be picked up and leave one for the opponent and win the game. So, 4 is also winning number.
Now, if there are 5 flags left, then no matter whichever number you choose among 1,2 or 3 you gonna leave other team with 4,3 or 2 respectively and all 3 are winning number. Therefore, 5 is loosing number.
Similarly, if you have 6,7,8 then by taking 1,2,3 respectively you can leaveyour opponent with number 5 and that is loosing number and you can win certainly. Therefore, 6,7,8 are also winning number.
Following above discussion, we see a pattern among the number that would make sure you win and with other numbers you loose.
we see after 1 if we go in positive direction on number line till 21 then every fourth number is loosing number because no matter what you choose you will leave your opponent with a winning number of flags and if your opponent knows and follows the optimal strategy can win. Therefore,
Loosing Numbers = 1,5,9,13,17 and 21
Winning Numbers = 2,3,4,6,7,8,10,11,12,14,15,16,18,19 and 20
a) STRATEGY
If you are a team that is stuck with any of loosing number that is 1,5,9,13,17 or 21 then no matter what strategy you follow you gonna loose. Therefore, there is n winning strategy in this situation given other team also knows the winning strategy
But if you have any of winning numbers at any point then you can follow the following strategy and can win
- 2 flags = pick up one flag and leave 1 for the opponent, win
- 3 flags = pick up 2 flags and leave 1 for the opponent, win
- 4 flags = pick up 3 flags and leave 1 for the opponent, win
- 6 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 1,2 or 3 based on the number of flags left and win
- 7 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 1,2 or 3 based on the number of flags left and win
- 8 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 1,2 or 3 based on the number of flags left and win
- 10 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 4,5 or 6 based on the number of flags left and win
- 11 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 4,5 or 6 based on the number of flags left and win
- 12 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 4,5 or 6 based on the number of flags left and win
- 14 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 7,8 or 9 based on the number of flags left and win
- 15 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 7,8 or 9 based on the number of flags left and win
- 16 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 7,8 or 9 based on the number of flags left and win
- 18 flags = pick up one flag, then no matter what your opponent chooses you follow either strategy number 10,11 or 12 based on the number of flags left and win
- 19 flags = pick up 2 flags, then no matter what your opponent chooses you follow either strategy number 10,11 or 12 based on the number of flags left and win
- 20 flags = pick up 3 flags, then no matter what your opponent chooses you follow either strategy number 10,11 or 12 based on the number of flags left and win
b)
Since team moving first will have 21 flags left on ground and it is a loosing number, no matter which number it chooses to pick it is gonna loose as it will leave other team with winning number. So, they will be indifferent between choosing 1,2 or 3
c)
If first teams removes 1 flag then best strategy for second team would be to remove 3 flags and again leave first team with loosing number
similarly, If first teams removes 2 flags then best strategy for second team would be to remove 2 flags and again leave first team with loosing number
and If first teams removes 3 flags then best strategy for second team would be to remove only 1 flag and again leave first team with loosing number
Which of the following steps should be completed first when simplifying ?
Answer:C
Step-by-step explanation:
MATH please heeeeeeeeeeeeeelp
Answer:
0, they all equal 0
Basic Math - the property of 0:
The multiplication property of zero says that zero times any number is equal to zero.
Step-by-step explanation:
Solution:
6(1/6)-2+1+12x0 = anything times 0 = 0
6(1/6)-2+1+0 = 0
6(1/6)-2+1 = 0
1-2+1 = 0
-1+1 = 0
=0
Which two expressions are equivalent?
A. 5x - 5 and 5(x - 5)
B. 5(x - 5) and 5x - 25
C. 5x - 1 and 5(x - 5)
D. 5x(x - 1) and 5x - 5
Answer:
B
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
if jessie had 3 feet of string and maxine had 34 inches of string, who had more string
Answer:
Jessie
Step-by-step explanation:
12 inches = 1 foot
Jessie has 3 feet of string. So, multiply 12 by 3.
12 × 3 = 36 inches
36 inches is greater than 34 inches. So, Jessie has more string.
Hope that helps.
What is the coordinate of the midpoint of DA?
D
Answer:
6
Step-by-step explanation:
Coordinate of point D = -6
Coordinate of point A = 18
The coordinate of the midpoint of DA = the coordinate of the point that is half the distance from D to A.
Distance of DA = |18 -(-6)| = |18 + 6| = 24
Midpoint = ½ the distance of DA = ½*24 = 12
Coordinate of the midpoint of DA = coordinate of D + 12 = -6 + 12 = 6
Or,
Coordinate of A - 12 = 18 - 12 = 6
Therefore, coordinate of the midpoint of DA = 6.
MO bisects LMN mLMN = 8x-23 and mLMO= 2x+37 solve for x and find LMN
Answer:
see explanation
Step-by-step explanation:
Since MO bisects ∠ LMN then
∠ LMO and ∠ NM0 are congruent, thus
∠ LMN = ∠ LMO + ∠ NMO , substitute values
8x - 23 = 2x + 37 + 2x + 37 = 4x + 74 ( subtract 4x from both sides )
4x - 23 = 74 ( add 23 to both sides )
4x = 97 ( divide both sides by 4 )
x = 24.25
Thus
∠ LMN = 8x - 23 = 8(24.25) - 23 = 194 - 23 = 171°
Determine whether the relation represents y as a function of x
2x+y^2=6
Answer:
rabi pati le crabby pati vi qui faste un fadi xx
What number is the opposite of 3.5
Answer:
-2/7
Step-by-step explanation: Thats the reciprocal so...
ezekiel has $250 in his bank account
The sum of two numbers is 50. The first number is 18 less than the 2nd number. What are the
2 numbers?
Answer:
2y + 2 + y = 50 (combine like terms)
3y + 2 = 50
3y = 50 - 2
3y = 48
y = 16
Substitute 16 for y in either equation 1 or 2 and solve for x (I chose equation 1):
x + 16 = 50
x = 50 - 16
x = 34
Step-by-step explanation:
what is 3/4 - (-4/5)
Answer: 1,55
Step-by-step explanation:
3/4 - (- 4/5) = 3/4 + 4/5 = (3x5)/20 + (4x4)/20 = 15/20 + 16/20 = 31/20 = 1,55
3/4 - (- 4/5) = 3/4 + 4/5 = 0,75 + 0.8 = 1,55
A movie theatre needed 980 popcorn
buckets. If each package has 9 buckets,
how
many packages will they need to
buy?
divide 5346 divided by 29
Answer:
184.3448276
Step-by-step explanation:
5346/29
184.3448276
Answer:
594
Step-by-step explanation:
HLEP MEE!!!!
Solve for Y!
Answer:
y = 10
Step-by-step explanation:
120 + 6y = 180
6y = 60
y = 10
Answer:
y = 10
Step-by-step explanation:
6y + 120 = 180
6y = 60
y = 10
If f(x) = 3x2 + 1 and g(x) = 1 – x, what is the value of (f – g)(2)?
12
14
36
38
Answer: 14
Step-by-step explanation:
Can you please help
Solve: x/2 = 7
Answers are:
x=5
X=7
X=9
X=14
Answer:
X = 14
14/2=7
Answer:
x=7
Step-by-step explanation:
you multiply a 2 to one already there and it eliminates the 2 entirely and then you get x=7
3|3t+1|=2|6t+3| Need hep
Answer:
your answer would be t=-1 or t=1
Step-by-step explanation:
-(2t-3) = 3t-2
Multiply
-2t+3 = 3t-2
Rearrange and Add up
-5t = -5
Divide both sides by 5
-t = -1
Multiply both sides by (-1)
t = 1
Which is the solution for the Negative Case
Solve the Positive Case
(2t-3) = 3t-2
Rearrange and Add up
-t = 1
Multiply both sides by (-1)
t = -1
have a great day!
If the temperature is -4 and rises by 12 , what is the new temperature ?
Answer:
-4 +12 = 8
the new temperature is 8 degree
Step-by-step explanation:
number 27 please.. i can not figure it out.
Answer: x=-12
Step-by-step explanation:
To solve for x, you want to use algebraic properties to isolate x.
[tex]\frac{x}{4}=2+\frac{x-3}{3}[/tex] [subtract both sides by [tex]\frac{x-3}{3}[/tex]]
[tex]\frac{x}{4}-\frac{x-3}{3}=2[/tex] [get common denominator]
[tex]\frac{3x}{12} -\frac{4x-12}{12} =2[/tex] [subtract]
[tex]\frac{-x+12}{12} =2[/tex] [multiply both sides by 12]
[tex]-x+12=24[/tex] [subtract both sides by 12]
[tex]-x=12[/tex] [divide both sides by -1]
[tex]x=-12[/tex]
Now, we know that x=-12.
Please help me solve this problem.
Answer:
h = V/πr²h = 10 inchesStep-by-step explanation:
V = πr²h1. Solving for h:
V = πr²hπr²h = Vπr²h/πr² = V/πr²h = V/πr²2. Finding the height of given cylinder, using formula above:
V = 90π, r = 3 inches, h = ?h = V/πr²h = 90π/π3²h = 10 inchesPaige's backyard is rectangular. It has a
length of 50 yards and a width of 80 feet.
Find the perimeter of Paige's backyard
using the correct number of significant
digits.
Answer:
probably 130
Step-by-step explanation:
well your gonna need to add both numbers together to see the out come of the answer
Simplify the expression. -6 + 3(2 - 4t) *
Answer:
-6 + 6 - 12t *
0 - 12t *
12t *
Therefore the answer is 12t *
joseph accepted a new job at a company with a contract guaranteeing annual raises. Joseph will get a raise of $5000 every year and had a starting salary of $70000. Write an equation for S, in terms of N, representing Josephs salary after working n years for the company.
Answer:S = 70,000 + 5000n
The equation that can be used to represent Josephs salary after working n years for the company is S = 70,000 + 5000n
How to write equation?Annual increase in salary = $5000Starting salary = $70,000S = Josephs salary after working n yearsn = number of yearsThe equation:
S = 70,000 + 5000(n)
S = 70,000 + 5000n
If n = 2
S = 70,000 + 5000n
= 70,000 + 5000(2)
= 70,000 + 10,000
= $80,000
Learn more about equation:
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Write this ratio as a fraction in lowest terms
60 minutes to 70 minutes
If the odds against Deborah's winning first prize in a chess tournament are 1 to 7, what is the probability of the event that she will win first prize?
Answer:
7/8
Step-by-step explanation:
Odds against are the ratio between losses and wins.
odds against = L / W
Probability of winning is the ratio between wins and all games.
probability = W / games
probability = W / (W + L)
The odds against are 1 to 7.
1/7 = L / W
W = 7 L
The probability is therefore:
P = (7 L) / (7 L + L)
P = 7/8
A hot air balloon has a height of 50 feet and is rising at a rate of 5 feet per minute. Another hot air balloon has a height of 70 feet and is rising at a rate of 3 feet per minute. After how many minutes will the balloons be at the same height?
If the number of minutes is represented by the variable x, then choose the equation that can be used to solve the problem.
Is the open sentence8−4x=4x true or false when x = 0? A.True B.False
Answer:
b.False it equals x=1
Simplifying a Fraction Expression
2/3 divided by (-4) - ( 1/6 - 8/6)
Answer:
Step-by-step explanation:
2/3 x -1/4 + 9/6)
-2/12 + 9/6
-1/6 + 8/6 = 7/6 = 1 1/6
As the manager of a store, you wish to determine the amount of money that people who visit this store are willing to spend on impulse buys on products placed near the checkout register. You sample twenty individuals and records their responses. Construct a frequency table for grouped data using five classes. 8,18,15,10,29,4,15,2,4,9,16,14,13,8,25,25,27,1,15,24
Answer:
The frequency table is listed down below.
Step-by-step explanation:
1:1
2:1
4:2
8:2
9:1
10:1
13:1
14:1
15:3
16:1
18:1
24:1
25:2
27:1
The frequency table for grouped data is shown in the picture and the classes are 1-6, 7-12, 13-18, 19-24, and 25-30
What are statistics?Statistics is a mathematical tool defined as the study of collecting data, analysis, understanding, representation, and organization. Statistics is described as the procedure of collecting data, classifying it, displaying that in a way that makes it easy to understand, and analyzing it even further.
We have:
The sample of twenty individuals and records their responses:
8,18,15,10,29,4,15,2,4,9,16,14,13,8,25,25,27,1,15,24
The classes are:
1-6, 7-12, 13-18, 19-24, and 25-30
The frequency table for grouped data is shown in the picture.
Thus, the frequency table for grouped data is shown in the picture and the classes are 1-6, 7-12, 13-18, 19-24, and 25-30
Learn more about the statistics here:
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Solve for x (see image)
Answer:
x = 18
Step-by-step explanation:
Since this is a right triangle, we can use the Pythagorean theorem
a^2 + b^2 = c^2
x^2 + 24^2 = ( 12+x)^2
FOIL the right side
x^2 +576 = 144 + 24x + x^2
Subtract x^2 from each side
576 = 144 + 24x
Subtract 144 from each side
576-144 = 24x
432 = 24x
Divide each side by 24
432/24 = 24x/24
18=x