1. **Problem statement:** Place one arithmetic operation (+, -, ×, ÷) between each pair of digits from 1 to 9 (digits can be rearranged and brackets used) to create an expression with the least possible positive integer value. 2. **Understanding the problem:** We have digits 1 through 9 and must insert operations between them to minimize the final positive integer result. We can rearrange digits and use brackets. 3. **Key insight:** To minimize a positive integer, we want to create a large denominator or subtract large values. Multiplication and division can drastically change values. 4. **Strategy:** Try to create a fraction with a large numerator and even larger denominator to get a small positive integer. 5. **Example approach:** Arrange digits as 1,2,3,4,5,6,7,8,9. Try expression: $$\frac{1}{2} + \frac{3}{4} + \frac{5}{6} + \frac{7}{8} + 9$$ Calculate approximate value: $$0.5 + 0.75 + 0.8333 + 0.875 + 9 = 11.9583$$ Not minimal. 6. **Better approach:** Try to create a fraction close to 1 or less. Try: $$\frac{1+2+3+4+5+6+7+8}{9} = \frac{36}{9} = 4$$ Positive integer 4. 7. **Try to get 1:** Try: $$\frac{1+2+3+4+5+6+7+8+9}{45}$$ Sum numerator: 45 Denominator: 45 Value: 1 8. **Is 1 the least positive integer possible?** Yes, since 0 or negative are not allowed. **Final answer:** The least possible positive integer value is **1**. **Summary:** By rearranging digits and using operations, the minimal positive integer achievable is 1.