1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alex_Xolod [135]
3 years ago
15

Sam colors each tile in a 4 by 4 grid white or black. A coloring is called rotationally

Mathematics
1 answer:
DiKsa [7]3 years ago
8 0

Answer:

  65,280

Step-by-step explanation:

Consider the 4×4 grid ...

  \left[\begin{array}{cc}a&b\\d&c\end{array}\right]

where each of a, b, c, d is a 2×2 array of tiles. Let's use the notation a' to represent the 2×2 array "a" rotated right 1/4 turn. For 90° rotational symmetry, we must have b=a', c=b'=a'', d=c'=b''=a'''. That is, once "a" is determined, the rest of the grid is determined. Since "a" consists of 4 tiles, each of which can be black or white, there are 2^4 = 16 patterns that have 90° rotational symmetry.

The same will be true of 270° rotational symmetry, for the same reason.

__

For 180° rotational symmetry, we must have c=a'' and d=b''. Then the combination of "a" and "b" together fully determines the grid. Together, "a" and "b" consist of 8 tiles, so there are 2^8 = 256 ways to pattern the grid so it will have 180° rotational symmetry. (Of those, 16 have 90° symmetry, and 16 have 270° symmetry. The sets are overlapping.)

__

The 16 tiles of the grid can be colored 2^16 = 65,536 different ways. As we have seen, 256 of those colorings result in 180° rotational symmetry. Then the number of colorings that have no rotational symmetry is ...

  65,536 -256 = 65,280 . . . . colorings not rotationally symmetric

You might be interested in
Given point A (3, 8) and point B (-5, 1). Find the following: a.) The coordinates of the midpoint of segment AB. Write your answ
ss7ja [257]

Answer:

Mid point:

= ( \frac{3 +  - 5}{2} , \:  \frac{1 + 8}{2} ) \\  = ( - 1, \: 4.5)

distance:

=  \sqrt{ {( - 5 - 3)}^{2} +  {(1 - 8)}^{2}  }  \\  =  \sqrt{64 + 49}  \\  =  \sqrt{113}  \\  = 10.6 \: units

slope:

=  \frac{1 - 8}{ - 5 - 3}  \\  =  \frac{ - 7}{ - 8}  \\  =  \frac{7}{8}

8 0
3 years ago
PLEASE HELP WILL MARK BRAINLIEST
jekas [21]

Answer:

10

Step-by-step explanation:

√81 = 9   (9+9-4*2)

4 * 2 = 8  (9+9-8)

9 + 9 = 18 (18 - 8)

18 - 8 = 10

7 0
2 years ago
Read 2 more answers
The lcm of 2 numbers is 60 and one of the numbers is 7 less than the other number
brilliants [131]
The two numbers are 5 and 12.
6 0
3 years ago
The total number of miles Jeremy walked on a hike can be represented by using the formula. If and n represents the total number
KatRina [158]

Answer:

A) 2 miles, 5 miles, 8 miles

Step-by-step explanation:

Let's get some details cleared.

We're interested in finding how many miles he walked at the end of each hour in a period of 3 hours.

We know what  is, so we can use that as our starting point.

Your answer is A.

7 0
2 years ago
Read 2 more answers
Solve -1 4/5 x= 9. <br> A) 5 <br> B) 1/5 <br> C) -5 <br> D) -1/5
Darina [25.2K]

Answer:

C

Step-by-step explanation:

Express - 1 \frac{4}{5} as an improper fraction, that is

- \frac{9}{5} x = 9 ( multiply both sides by 5 )

- 9x = 45 ( divide both sides by - 9 )

x = - 5 → C

7 0
3 years ago
Other questions:
  • Rewrite 3 1 over 5<br> as a mixed number?
    6·2 answers
  • Which is a zero of the quadratic function f(x) = 4x2 + 24x + 11?
    10·2 answers
  • Examples of when you will use box and whisker plots in the real world
    11·1 answer
  • What is the value of x in the equation 0.7 x minus 1.4 equals -3.5 ​
    7·2 answers
  • Can you check my math? If I did something wrong could you explain why.
    11·1 answer
  • On a circular playground, the distance from its center to the edge of the playground is 36 feet. What is the approximate circumf
    10·1 answer
  • What is 1 fifth of a clock
    6·2 answers
  • Find the domain of the function. (Enter your answer using interval notation.) f(x) = 7 − 1 2 x if x ≤ 2 8x − 4 if x &gt; 2 g
    10·1 answer
  • maxwell borrows 1,000 to repair his car at 9% interest if he pays the loan back over the next 1.5 years how much does he pay bac
    7·2 answers
  • ..........................................................................................................
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!