Mathematical Puzzles

Using Excel, the numbers are: 616, 637, 677 and 696. This sums to 2626. So the answer is 626.
/* I know this approach was not what you were looking for, but can’t resist using Excel when I have a spreadsheet open. 🙂 */

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Continue reading about Consider all three-digit numbers that are greater than the sum of the squares of their digits by exactly 543. What are the last three digits of the sum of these numbers?

Amit BhatnagarThis is mathematical proof, but I guess this is fairly intuitive too..  Line up the series of numbers from 1 to (n-1) once, and in reverse right below that:    1    +     2 + &n…

Continue reading about What is an intuitive explanation for why (n choose 2) happens to be the sum of integers from 1 to n-1?

Praveen Reddy Vaka and Anon-User have already demonstrated two mathematically correct ways of handling this. I will discuss a slightly less accurate, but a reasonably practical and relatively easy approach using Excel:

  1. Let the four square corners of the square sit at (0,0), (0,1), (1,0) and (1,1). So, the square center A will always be at (0.5,0.5). Now, take two random points by generating coordinates using Excel’s rand() function, which generates a random number between 0 and 1, hence guaranteeing the coordinate to be within the square.
  2. Calculate the radius r of the circle by dividing the distance between Point1 and Point2 by 2
  3. Locate the center O of the circle by calculating the midpoint of Point1 and Point2.
  4. Calculate the length of OA, i.e. the distance between square center and circle center
  5. Define Success: Center of the square is with in the circle (r > OA). Assign 1 to success and 0 to failure
  6. Repeat the process a large number of times in excel and estimate the probability by dividing success count by iteration count.
  7. Press F9 a few times to refresh the data, and you will see that the probability is around 0.5
  8. (Optional) If you have access to any advanced too like @Risk from Palisade, you can automate step 6 for a number of times, and graphically see the probability converging around 0.5

A screenshot of my excel solution for this is included below:

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Continue reading about What is the probability of choosing two points inside a square such that center of the square lies in the circle formed by taking the points as diameter?