The problem states that there is a set of 10 cards, each showing one of the digits from 0 to 9. These cards are divided between five envelopes such that there are exactly two cards in each envelope. The sum of the digits inside each envelope is written on the envelope.

The given sums on the envelopes are:

• 4
• 5
• 10
• 13
• 13

We need to determine what pair of cards is definitely in an envelope marked "13."

Step 1: Identify possible pairs for each sum

• Sum = 4: Possible pairs (0,4), (1,3)
• Sum = 5: Possible pairs (0,5), (1,4), (2,3)
• Sum = 10: Possible pairs (1,9), (2,8), (3,7), (4,6)
• Sum = 13: Possible pairs (4,9), (5,8), (6,7)

Step 2: Consider all pairs

Since the total of all digits from 0 to 9 is $0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$, the sum of all the numbers must equal the sum of the digits in all envelopes. The sums provided add up to $4 + 5 + 10 + 13 + 13 = 45$, which is correct.

Step 3: Analyze the pairs to find the pair that must be in the envelope marked "13"

1. If we place (6,7) in one envelope marked "13", we cannot use them for any other sum, so pairs (4,9) and (5,8) must be split between the other envelope marked "13" and the envelope marked "10".

2. Let's assume (6,7) is in the first "13" envelope:

   • For the other "13", the remaining pairs are (4,9) and (5,8).
   • If (4,9) is used for the second "13" envelope, then (5,8) must be used in the "10" envelope.
   • But this leaves no valid pairs for the "4" and "5" envelopes.

3. Next, assume (5,8) is in one of the "13" envelopes:

   • Then (4,9) must be in the other "13".
   • This leaves (6,7) for the "10" envelope.

4. This leaves the pairs (1,3) for the "4" and (0,5) for the "5", which fit perfectly.

Conclusion

Thus, the pair (4,9) must definitely be in one of the envelopes marked "13".

Would you like more details or have any other questions?

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Relative Questions:

1. What are the possible pairs for an envelope marked with the sum "15"?
2. How would the solution change if there was only one envelope marked with "13"?
3. Can the solution be generalized for any set of numbers from 0 to n?
4. What is the mathematical strategy to determine pairings in such problems?
5. What if the envelopes had three cards each instead of two?

Tip: When solving such problems, listing all possible pairs and considering all constraints systematically can help identify the correct configuration.