Final answer:
The question involves finding sets of two distinct numbers whose sum equals the given square numbers without any repetition of the numbers in the sets. After testing various pairings, a feasible solution is identified where each line's sum yields the required square number and each number within the pair is unique.
Step-by-step explanation:
The question asks to place four different numbers in 'files' (likely meaning slots or positions) so that the sum of the numbers at the end of each line gives a perfect square number. To find such numbers, we must consider pairs of numbers that add up to each of the given square numbers. For example, 16 can be achieved by 9 + 7, 25 by 16 + 9, 36 by 25 + 11, and 49 by 36 + 13. Let's say we take 9 as a common number in the first two sums, the breakdown may look like this:
- 16 = 9 + 7
- 25 = 16 + 9
- 36 = 25 + 11
- 49 = 36 + 13
However, we see that the chosen numbers are not all distinct. Let's revise the strategy. Since the question requires the numbers at the ends of the lines to be different, we can pick numbers that add up to these squares in such a way. Here's a possible solution:
- Line 1: 16 = 10 + 6
- Line 2: 25 = 17 + 8
- Line 3: 36 = 13 + 23
- Line 4: 49 = 28 + 21
Notice that none of the numbers used are repeated, and this satisfies the condition of the problem. This is just one example, and different combinations can be used as long as the sum is a square number and the numbers in the 'files' are different.