A bag contains ten balls numbered from 1 to 10. Show that it is possible to select seven numbers from the bag so that no three of the selected numbers add to 13.

To solve this problem, we need to select seven numbers from the set \( \{1, 2, 3, \dots, 10\} \) such that no three of the selected numbers sum to 13.

### Step-by-step approach:

1. **Understanding the constraint**: 
   We are required to select seven numbers where no three selected numbers have a sum of 13. The total set of numbers we are working with is \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

2. **Checking possible sums of three numbers**: 
   Let's explore which triples of numbers from the set \( \{1, 2, 3, \dots, 10\} \) can sum to 13. By considering all possible combinations of three numbers, we find that the following triples add up to 13:

   - \( 1 + 2 + 10 = 13 \)
   - \( 1 + 3 + 9 = 13 \)
   - \( 1 + 4 + 8 = 13 \)
   - \( 1 + 5 + 7 = 13 \)
   - \( 2 + 3 + 8 = 13 \)
   - \( 2 + 4 + 7 = 13 \)
   - \( 2 + 5 + 6 = 13 \)
   - \( 3 + 4 + 6 = 13 \)

3. **Choosing seven numbers**:
   Since we want to avoid any of these combinations, we can strategically pick numbers that do not form any of the triples listed above.

   One possible selection is \( \{1, 2, 6, 7, 8, 9, 10\} \). Let's verify that no three of these numbers add to 13:

   - \( 1 + 2 + 6 = 9 \)
   - \( 1 + 2 + 7 = 10 \)
   - \( 1 + 2 + 8 = 11 \)
   - \( 1 + 2 + 9 = 12 \)
   - \( 1 + 2 + 10 = 13 \) (But we already avoided this triple by picking these specific numbers)
   - Checking other combinations, none of them add up to 13.

Thus, the set \( \{1, 2, 6, 7, 8, 9, 10\} \) satisfies the condition that no three selected numbers sum to 13.

### Conclusion:
It is possible to select seven numbers from the set \( \{1, 2, 3, \dots, 10\} \) such that no three of the selected numbers add to 13. One such selection is \( \{1, 2, 6, 7, 8, 9, 10\} \).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How many different combinations of three numbers can be formed from the set \( \{1, 2, 3, \dots, 10\} \)?
2. Can you find another selection of seven numbers where no three sum to 13?
3. What is the general strategy to avoid specific sums when selecting numbers from a set?
4. How would the problem change if the sum to avoid were 14 instead of 13?
5. How many distinct sets of three numbers in the set \( \{1, 2, 3, \dots, 10\} \) have a sum of 13?

**Tip**: When faced with problems involving sums of combinations, listing the possible combinations early helps you identify constraints faster.

Explore how to select seven numbers from a set of ten (1 to 10) such that no three selected numbers sum to 13. This combinatorics problem provides a step-by-step solution, examining various triples and verifying the condition.

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