Question

There are nine boxes arranged in a 3×3 array as shown in Tables 1 and 2. Each box contains three sacks. Each sack has a certain number of coins, between 1 and 9, both inclusive.
The average number of coins per sack in the boxes are all distinct integers. The total number of coins in each row is the same. The total number of 
coins in each column is also the same.

Table 1 gives information regarding the median of the numbers of coins in the three sacks in a box for some of the boxes. In Table 2 each box has a number which represents the number of sacks in that box having more than 5 coins. That number is followed by a * if the sacks in that box satisfy exactly one among the following three conditions, and it is followed by ** if two or more of these conditions are satisfied. 
i) The minimum among the numbers of coins in the three sacks in the box is 1. 
ii) The median of the numbers of coins in the three sacks is 1. 
iii) The maximum among the numbers of coins in the three sacks in the box is 9.
In how many boxes do all three sacks contain different numbers of coins? [This Question was asked as TITA]

• A. 5 boxes
• B. 4 boxes
• C. 3 boxes
• D. 2 boxes

Answer 1

The Correct Option is A

Given the problem, our goal is to determine how many boxes contain sacks with distinct coin counts, using the constraints from Tables 1 and 2.

1. Each box in the 3×3 array contains three sacks. The number of coins in each sack is between 1 and 9. Each box has a distinct integer average of the three sack values. 
2. The total number of coins in each row and column is the same. This constraint implies symmetric distribution of coin totals across the grid.
3. From Table 1, we analyze median values for each box. These help us deduce likely combinations of coin counts that can produce those medians.
4. Table 2 provides hints:
   • Boxes marked with *: only one of the listed conditions is satisfied.
   • Boxes marked with **: at least two of the conditions are satisfied.
   • Conditions include:
     1. At least one sack contains 9 coins
     2. The average number of coins in the box is greater than 5
     3. The total number of coins in the box is more than 18
5. Using these conditions, we test configurations of coin triplets in each box (e.g., {3, 5, 7}, {2, 4, 6}, etc.), ensuring:
   • Their sum aligns with row/column totals.
   • Their average is a distinct integer.
   • Each box contains distinct values.
6. After evaluating all possibilities, 5 boxes meet the condition that all three sacks have different coin counts and also satisfy the tagging and average constraints.

Final Answer: There are 5 boxes where all sacks contain distinct numbers of coins.