1. **Problem statement:** We have six cards numbered 1, 2, 3, 5, 7, and 9. We want to fill the blanks in the expression $$\square . \square \square + \square \square . \square$$ using each card exactly once to make the sum as small as possible. 2. **Understanding the problem:** - The first number has 1 digit before the decimal and 2 digits after: $a.bc$ - The second number has 2 digits before the decimal and 1 digit after: $de.f$ - All digits $a,b,c,d,e,f$ must be distinct and chosen from {1,2,3,5,7,9}. - We want to minimize the sum $a.bc + de.f$. 3. **Strategy:** - To minimize the sum, minimize the larger number first (the two-digit number $de.f$), then minimize the smaller number $a.bc$. - The two-digit number $de.f$ is larger than the one-digit number $a.bc$ because it has two digits before the decimal. 4. **Step-by-step solution:** - Assign the smallest two digits to $d$ and $e$ to minimize $de.f$. - The smallest digits are 1, 2, 3, 5, 7, 9. - The smallest two-digit number from these digits is 12 (digits 1 and 2). - For the decimal digit $f$ of the second number, choose the smallest remaining digit after 1 and 2 are used. - Remaining digits after using 1 and 2: 3, 5, 7, 9. - Choose $f=3$ to minimize $de.f = 12.3$. - Now assign digits to the first number $a.bc$ from remaining digits {5,7,9}. - To minimize $a.bc$, assign the smallest digit to $a$, then next smallest to $b$, then next to $c$. - So $a=5$, $b=7$, $c=9$. - First number is $5.79$. 5. **Calculate the sum:** $$5.79 + 12.3 = 18.09$$ 6. **Check if swapping digits in the first number can reduce sum:** - If we try $a=3$, but 3 is used in second number decimal digit. - If we try $a=3$ in first number, then second number decimal digit must be from {5,7,9}, which is larger than 3. - So $12.3$ is minimal for second number. 7. **Try swapping decimal digits in first number:** - If $a=5$, $b=9$, $c=7$, first number is $5.97$. - Sum: $5.97 + 12.3 = 18.27$ (larger). 8. **Try swapping decimal digit in second number:** - If $f=5$, second number is $12.5$. - Remaining digits for first number: {3,7,9}. - First number minimal is $3.79$. - Sum: $3.79 + 12.5 = 16.29$ (smaller than 18.09). 9. **Try $f=7$:** - Second number: $12.7$. - First number digits: {3,5,9}. - First number minimal: $3.59$. - Sum: $3.59 + 12.7 = 16.29$ (same as before). 10. **Try $f=9$:** - Second number: $12.9$. - First number digits: {3,5,7}. - First number minimal: $3.57$. - Sum: $3.57 + 12.9 = 16.47$ (larger). 11. **Try other two-digit numbers for second number:** - Next smallest two-digit number is 13. - If $de=13$, $f=2$ (smallest remaining digit). - Second number: $13.2$. - First number digits: {5,7,9}. - First number minimal: $5.79$. - Sum: $5.79 + 13.2 = 18.99$ (larger). 12. **Try $de=15$, $f=2$:** - Second number: $15.2$. - First number digits: {3,7,9}. - First number minimal: $3.79$. - Sum: $3.79 + 15.2 = 18.99$ (larger). 13. **Conclusion:** - The minimal sum found is $16.29$ with two possible assignments: - $5.79 + 12.5 = 16.29$ - $3.59 + 12.7 = 16.29$ 14. **Final answer:** $$\boxed{16.29}$$