1. **Problem Statement:** Use the numbers 25, 8, 100, 2, 75, and 50 exactly once each with any operations to reach the target value 431. 2. **Understanding the Rules:** - Each number must be used exactly once. - Operations (+, -, *, /) can be used multiple times. - The goal is to combine these numbers to get exactly 431. 3. **Approach:** - Start by considering combinations of numbers that sum or multiply close to 431. - Use addition and subtraction to adjust the total. 4. **Example Solution:** - Calculate $$100 \times 4 = 400$$ (using 100 and 2 twice is not allowed, so instead use 100 and 2 to get 200, then multiply by 2 again is invalid, so try another way) - Instead, try $$75 + 50 = 125$$ - Then $$125 \times 2 = 250$$ - Add 100: $$250 + 100 = 350$$ - Add 25: $$350 + 25 = 375$$ - Add 8: $$375 + 8 = 383$$ - We need 431, so try multiplying 8 by something else or rearranging. 5. **Correct Calculation:** - Multiply 8 and 50: $$8 \times 50 = 400$$ - Add 75: $$400 + 75 = 475$$ - Subtract 25: $$475 - 25 = 450$$ - Subtract 100: $$450 - 100 = 350$$ - Add 2: $$350 + 2 = 352$$ (Not 431, so try another way) 6. **Final Working Solution:** - Multiply 8 and 50: $$8 \times 50 = 400$$ - Add 75: $$400 + 75 = 475$$ - Subtract 25: $$475 - 25 = 450$$ - Subtract 100: $$450 - 100 = 350$$ - Add 2: $$350 + 2 = 352$$ (Still not 431, so try another approach) 7. **Alternative Solution:** - Multiply 75 and 2: $$75 \times 2 = 150$$ - Add 100: $$150 + 100 = 250$$ - Multiply 8 and 25: $$8 \times 25 = 200$$ - Add 250 and 200: $$250 + 200 = 450$$ - Subtract 50: $$450 - 50 = 400$$ (Still not 431) 8. **Try using division:** - Divide 100 by 2: $$100 \div 2 = 50$$ - Add 50: $$50 + 50 = 100$$ - Multiply 8 and 25: $$8 \times 25 = 200$$ - Add 100 and 200: $$100 + 200 = 300$$ - Add 75: $$300 + 75 = 375$$ - Add 50: $$375 + 50 = 425$$ (Close to 431) 9. **Adjusting to 431:** - Instead of adding 50, add 8: $$375 + 8 = 383$$ - Need 48 more, try multiplying 2 and 25: $$2 \times 25 = 50$$ - Add 50 to 383: $$383 + 50 = 433$$ (Too high) 10. **Conclusion:** - The problem requires creative use of operations and numbers. - The key is to try different combinations and operations to reach exactly 431. **Final note:** This problem is a challenging puzzle that encourages practice with arithmetic operations and number manipulation.