answer:Let's denote the number of participants as n. Each participant plays with every other participant exactly once. So, the first participant will play n-1 games (with each of the other participants), the second participant will also play n-1 games, but we have already counted the game with the first participant, so we count n-2 additional games for the second participant, and so on. The total number of games played is the sum of the series: (n-1) + (n-2) + (n-3) + ... + 3 + 2 + 1 This is the sum of an arithmetic series with n-1 terms, and the sum of such a series can be calculated with the formula: Sum = (number of terms / 2) * (first term + last term) In this case, the number of terms is n-1, the first term is 1, and the last term is n-1. So the sum is: Sum = ((n-1) / 2) * (1 + (n-1)) We know that the sum is 171, so we can set up the equation: 171 = ((n-1) / 2) * (n) Multiplying both sides by 2 to get rid of the fraction gives us: 342 = (n-1) * n Expanding the right side gives us a quadratic equation: n^2 - n - 342 = 0 To solve this quadratic equation, we can use the quadratic formula or factor it if possible. Let's see if we can factor it: We are looking for two numbers that multiply to -342 and add up to -1. These numbers are -19 and 18 because -19 * 18 = -342 and -19 + 18 = -1. So we can factor the equation as: (n - 19)(n + 18) = 0 This gives us two possible solutions for n: n - 19 = 0 or n + 18 = 0 n = 19 or n = -18 Since the number of participants cannot be negative, we discard n = -18. Therefore, the number of participants in the tournament is boxed{19} .

answer:Let's calculate the total number of legs for the animals we know the quantity of: Dogs: 5 dogs * 4 legs each = 20 legs Cats: 7 cats * 4 legs each = 28 legs Turtles: 3 turtles * 4 legs each = 12 legs Goat: 1 goat * 4 legs = 4 legs Now, let's add up the legs of these known animals: 20 (dogs) + 28 (cats) + 12 (turtles) + 4 (goat) = 64 legs We know that all the animals together have 72 legs, so let's subtract the legs of the known animals from the total: 72 total legs - 64 legs (known animals) = 8 legs Since horses have 4 legs each, we can divide the remaining legs by 4 to find out how many horses Daniel has: 8 legs / 4 legs per horse = 2 horses Daniel has boxed{2} horses.

answer:To solve the problem of determining the minimum number of colors required to paint a 3 times 3 grid such that no two squares of the same color appear in the same row, column, or diagonal, we proceed as follows: Step 1: Assign colors to the main diagonal We begin by assigning the colors 1, 2, and 3 to the main diagonal elements of the grid. The grid looks like this: [ begin{array}{|c|c|c|} hline 1 & & X hline & 2 & hline Y & & 3 hline end{array} ] Step 2: Evaluate the elements on the secondary diagonal Let's consider the colors for the squares at the ends of the secondary diagonal, labeled X and Y. 1. The color of X: - X is in the same row as color 1. - X is in the same column as color 3. - X is in the same diagonal as color 2. Consequently, X cannot be 1, 2, or 3; thus, X must be a new color (say color 4). 2. The color of Y: - Y is in the same row as color 3. - Y is in the same column as color 1. - Y is in the same diagonal as color 2. Similarly, Y must also be a new color (say color 5). [ begin{array}{|c|c|c|} hline 1 & & 4 hline & 2 & hline 5 & & 3 hline end{array} ] Step 3: Ensuring distinct colors in the remaining squares The remaining squares must be filled with colors different from those already used in their respective rows, columns, and diagonals. Verify and assign suitable colors: [ begin{array}{|c|c|c|} hline 1 & 3 & 4 hline 3 & 2 & 1 hline 5 & 1 & 3 hline end{array} ] Verification 1. **Rows:** - Row 1: 1, 3, 4 - Row 2: 3, 2, 1 - Row 3: 5, 1, 3 2. **Columns:** - Column 1: 1, 3, 5 - Column 2: 3, 2, 1 - Column 3: 4, 1, 3 3. **Diagonals:** - Main Diagonal: 1, 2, 3 - Secondary Diagonal: 4, 2, 5 Each row, column, and diagonal contains distinct colors, and hence the coloring satisfies the given constraints. Conclusion Thus, the minimum number of colors necessary to paint the 3 times 3 grid such that no two squares of the same color appear in the same row, column, or diagonal is: [ boxed{5} ]

answer:1. **Understand the Problem**: Given that person A and person B agree to meet within 10 days at a certain location, we need to determine the probability that they will meet. It's given that one person waits for 3 days after their arrival before they leave if the other person has not yet arrived. If they arrive on a day uniformly chosen from 0 to 10, we need to find the probability that the absolute difference between their arrival days is no more than 3. 2. **Define Variables**: Let ( x ) and ( y ) represent the days on which persons A and B arrive respectively. Then ( 0 leqslant x leqslant 10 ) and ( 0 leqslant y leqslant 10 ). 3. **Set the Condition for Meeting**: They will meet if ( |x - y| leqslant 3 ). 4. **Graphical Solution**: - We can visualize this problem using a coordinate system where ( (x, y) ) lies within the square defined by ( 0 leqslant x leqslant 10 ) and ( 0 leqslant y leqslant 10 ). - The condition ( |x - y| leqslant 3 ) describes a band around the line ( x = y ) where the width of the band is 6 units. 5. **Calculate the Area Where They Meet**: - The square ( OABC ) where the coordinates lie has an area of ( 100 ) (since it's a ( 10 times 10 ) square). - The region where ( |x - y| leqslant 3 ) can be found by understanding it includes two lines ( y = x + 3 ) and ( y = x - 3 ) within this square. - Let's find the area of the region where ( x ) is within ( y pm 3 ): - Visualizing the problem geometrically, this band forms a central band around the diagonal from ( (0, 0) ) to ( (10, 10) ). 6. **Calculate the Effective Area**: - The total possible area for both arrivals is ( 10 times 10 = 100 ). - The area not covered by the overlap region consists of two triangles (above ( y = x + 3 ) and below ( y = x - 3 )). Each triangle has leg lengths of 7 units, so the area of each triangle is ( frac{1}{2} times 7 times 7 = 24.5 ). - Since there are two such triangles, the total area outside the region where they meet is ( 2 times 24.5 = 49 ). 7. **Meeting Probability**: - The area where they will meet is ( 100 - 49 = 51 ). Thus, the probability ( P ) that they will meet is: [ P = frac{text{Favorable Area}}{text{Total Area}} = frac{51}{100} ] # Conclusion: [ boxed{frac{51}{100}} ]