Problem 5.4. Arrange the digits from 1 to 6 (each must be used exactly once) so that the sum of the three numbers located on each of the 7 lines is equal to 15. In your answer, indicate which digits should be placed at positions $A-F$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a7569897979ba1659de3g-09.jpg?height=363&width=456&top_left_y=612&top_left_x=499)

(a) Instead of the letter $A$

(b) Instead of the letter $B$

(c) Instead of the letter $C$

(d) Instead of the letter $D$

(e) Instead of the letter $E$

(f) Instead of the letter $F$
(1) the digit 1 should be placed

(2) the digit 2 should be placed

(3) the digit 3 should be placed

(4) the digit 4 should be placed

(5) the digit 5 should be placed

(6) the digit 6 should be placed
Answer: $A=4, B=1, C=2, D=5, E=6, F=3$.

Solution. According to the condition, $A, D, E$ are distinct digits, not exceeding 6, the sum of which is 15. If these digits are taken to be the maximum possible, then their sum is $4+5+6=15$. Therefore, $A, D, E$ are $4,5,6$ in some order (if at least one of the digits is no more than 3, then the sum of all three digits is no more than $3+5+6=14$) and $D \neq 6$ (otherwise $B+D+9>15$). Hence, $E=6$.

Since $7+C+E=15$ and $E=6$, we get $C=2$.

Since $9+C+A=15$ and $C=2$, we get $A=4$.

Since $A+8+F=15$ and $A=4$, we get $F=3$.

Since $7+D+F=15$ and $F=3$, we get $D=5$.

Since $9+D+B=15$ and $D=5$, we get $B=1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a7569897979ba1659de3g-10.jpg?height=361&width=456&top_left_y=90&top_left_x=499)

It is easy to verify that the obtained arrangement satisfies all the conditions.