What is the triangular pattern that you are asked to arrange if it is a 3 square arrangements and squares?

Steps to Solve

1. Understanding the problems
   The problem consists of finding certain values based on the given questions, which involve square and triangular numbers.

2. Finding the square of certain values
   Starting with question 1, we need to find the squares of the expected values. The largest expected square below 78 must be calculated. The largest square less than 78 is (8^2 = 64).

3. Finding triangular numbers
   For question 2, a triangular number can be found using the formula for the (n)-th triangular number:
   $$ T_n = \frac{n(n + 1)}{2} $$
   We search for the largest (n) such that (T_n < 78). By trial with different (n), we find (T_{12} = \frac{12(13)}{2} = 78) is not valid since it is equal; thus, we take (T_{11} = 66).

4. Accessing additional triangular numbers
   Question 3 requires us to find the (n)-th triangular number where (n = 12). Using the triangular number formula:
   $$ T_{12} = \frac{12(12 + 1)}{2} = \frac{12 \times 13}{2} = 78 $$

5. Finding consecutive triangular numbers
   Question 4 asks for two consecutive triangular numbers. The values we found so far are 66 (for (n=11)) and 78 (for (n=12)). Thus, the consecutive triangular numbers are (T_{11} = 66) and (T_{12} = 78).

6. Finding the square and the arrangement of numbers
   Finally, question 5 asks to find (15) squares arranged in a square pattern. The total number of squares must therefore be (3 \times 3 = 9) since (3^2 < 15).