💡Okay, so I have this problem about symmetric scanning codes. It's a 9x9 grid where each square can be black or white. The code is called symmetric if it looks the same after a 180-degree rotation around its center. I need to find the total number of possible symmetric scanning codes, making sure there's at least one black and one white square.Hmm, let me start by understanding what 180-degree rotational symmetry means. If I rotate the grid 180 degrees, every square should end up in the position of another square, and the color should remain the same. So, for each square, there's another square that it maps to after the rotation. Except for the center square, which maps to itself because it's in the middle.So, in a 9x9 grid, the center is the 5th row and 5th column. That square is alone in its symmetry pair. All other squares have a unique pair that they rotate into. So, how many such pairs are there?Let me visualize the grid. Each square (i,j) will map to (10-i,10-j) because it's a 9x9 grid, so rows and columns go from 1 to 9. So, for example, the square at (1,1) maps to (9,9), (1,2) maps to (9,8), and so on.So, to count the number of independent squares, I need to figure out how many unique pairs there are plus the center square. Since the grid is 9x9, there are 81 squares in total. The center square is one, so the remaining 80 squares form 40 pairs. Therefore, the number of independent squares is 40 + 1 = 41.Each of these independent squares can be either black or white. So, for each of the 41 squares, there are 2 choices. That would give me 2^41 possible colorings. However, the problem states that there must be at least one black and one white square. So, I need to subtract the cases where all squares are black or all are white.There are two such cases: all black and all white. So, subtracting these two from the total gives 2^41 - 2.Wait, let me double-check. Each pair must have the same color, so choosing the color for one square in the pair determines the color of the other. Since there are 40 pairs and 1 center square, that's 41 independent choices. Each choice has 2 options, so 2^41. Then subtract the two monochromatic cases. That makes sense.Is there any other constraint I'm missing? The problem only specifies that there must be at least one of each color, so I think that's it. So, the total number of symmetric scanning codes is 2^41 - 2.I think that's the answer. Let me just recap:- 9x9 grid, 81 squares.- 180-degree rotational symmetry means each square is paired with another, except the center.- 40 pairs + 1 center = 41 independent squares.- Each can be black or white: 2^41.- Subtract all-black and all-white: 2^41 - 2.Yeah, that seems right. I don't see any mistakes in my reasoning.

💡First, determine the total number of blue tickets needed for 10 yellow tickets.Each yellow ticket requires 10 red tickets, so for 10 yellow tickets:10 yellow tickets × 10 red tickets/yellow ticket = 100 red ticketsEach red ticket requires 10 blue tickets, so for 100 red tickets:100 red tickets × 10 blue tickets/red ticket = 1000 blue ticketsTom currently has 8 yellow tickets, 3 red tickets, and an unknown number of blue tickets. Convert these into blue tickets:For 8 yellow tickets:8 yellow tickets × 10 red tickets/yellow ticket × 10 blue tickets/red ticket = 800 blue ticketsFor 3 red tickets:3 red tickets × 10 blue tickets/red ticket = 30 blue ticketsTom needs 163 more blue tickets to reach the total of 1000 blue tickets. Add the blue tickets he needs to the ones he already has:800 (from yellow) + 30 (from red) + 163 (needed) = 993 blue ticketsSubtract this from the total needed to find the current number of blue tickets:1000 (total needed) - 993 (current equivalent in blue tickets) = 7 blue tickets

💡First, I need to solve the inequality ( x + 1 < frac{4 + 3x}{2} ).To eliminate the fraction, I'll multiply both sides of the inequality by 2:[ 2(x + 1) < 4 + 3x ]Next, I'll distribute the 2 on the left side:[ 2x + 2 < 4 + 3x ]Now, I'll move all terms containing ( x ) to one side and constants to the other side. I'll subtract ( 2x ) from both sides:[ 2 < 4 + x ]Then, I'll subtract 4 from both sides to isolate ( x ):[ -2 < x ]This can also be written as:[ x > -2 ]So, the solution set of the inequality is all real numbers greater than -2.

💡To convert 3.36 into a simplified fraction, I start by expressing it as the sum of the whole number and the decimal part: 3 + 0.36.Next, I convert the decimal 0.36 into a fraction by writing it as 36 over 100, which is 36/100.To simplify 36/100, I find the greatest common divisor of 36 and 100, which is 4. Dividing both the numerator and the denominator by 4, I get 9/25.Finally, I combine the whole number 3 with the simplified fraction 9/25. Converting 3 into a fraction with the same denominator, I have 75/25. Adding 75/25 and 9/25 gives me 84/25.Therefore, 3.36 as a simplified fraction is 84/25.

💡First, I need to evaluate the expression (-|-1|).I start by calculating the absolute value of (-1), which is (|-1| = 1).Next, I apply the negative sign to the result of the absolute value: (-1).Therefore, the value of the expression (-|-1|) is (-1).

💡First, identify the total number of ways to select three different numbers from the set ({1, 2, 3, 4, 5}). This is calculated using the combination formula:[binom{5}{3} = 10]Next, determine which of these combinations form Pythagorean triples. A Pythagorean triple consists of three positive integers (a), (b), and (c) such that (a^2 + b^2 = c^2). Check each combination:- (1, 2, 3): (1^2 + 2^2 = 1 + 4 = 5 neq 3^2 = 9)- (1, 2, 4): (1^2 + 2^2 = 1 + 4 = 5 neq 4^2 = 16)- (1, 2, 5): (1^2 + 2^2 = 1 + 4 = 5 neq 5^2 = 25)- (1, 3, 4): (1^2 + 3^2 = 1 + 9 = 10 neq 4^2 = 16)- (1, 3, 5): (1^2 + 3^2 = 1 + 9 = 10 neq 5^2 = 25)- (1, 4, 5): (1^2 + 4^2 = 1 + 16 = 17 neq 5^2 = 25)- (2, 3, 4): (2^2 + 3^2 = 4 + 9 = 13 neq 4^2 = 16)- (2, 3, 5): (2^2 + 3^2 = 4 + 9 = 13 neq 5^2 = 25)- (2, 4, 5): (2^2 + 4^2 = 4 + 16 = 20 neq 5^2 = 25)- (3, 4, 5): (3^2 + 4^2 = 9 + 16 = 25 = 5^2 = 25)Only the combination (3, 4, 5) satisfies the Pythagorean theorem.Finally, calculate the probability by dividing the number of successful outcomes by the total number of possible outcomes:[text{Probability} = frac{1}{10}]