1. **Problem Statement:** Place the digits 1 to 6 in a magic triangle so that each side of the triangle sums to the given number. We have two cases: (i) sum = 9 (ii) sum = 10 2. **Understanding the Magic Triangle:** Each side has 3 digits, and the sum of these digits must equal the target sum. 3. **Variables:** Let the triangle vertices be $A$, $B$, and $C$, and the midpoints on each side be $D$, $E$, and $F$. The sides are: - Side 1: $A + D + B$ - Side 2: $B + E + C$ - Side 3: $C + F + A$ All digits 1 to 6 must be used exactly once. --- ### Case (i): sum = 9 4. **Set up equations:** $$ A + D + B = 9 \\ B + E + C = 9 \\ C + F + A = 9 $$ 5. **Sum all three equations:** $$ (A + D + B) + (B + E + C) + (C + F + A) = 3 \times 9 = 27 $$ 6. **Group terms:** $$ 2(A + B + C) + (D + E + F) = 27 $$ 7. **Sum of digits 1 to 6:** $$ A + B + C + D + E + F = 1 + 2 + 3 + 4 + 5 + 6 = 21 $$ 8. **From step 6 and 7, substitute $D + E + F = 21 - (A + B + C)$:** $$ 2(A + B + C) + 21 - (A + B + C) = 27 \\ (A + B + C) + 21 = 27 \\ A + B + C = 6 $$ 9. **Therefore:** $$ D + E + F = 21 - 6 = 15 $$ 10. **Find triples $A,B,C$ with sum 6 from digits 1 to 6:** Possible triples (distinct digits): - (1,2,3) 11. **Assign $A,B,C = 1,2,3$ in some order and $D,E,F = 4,5,6$ in some order.** 12. **Check side sums:** For side 1: $A + D + B = 9$ implies $D = 9 - (A + B)$. Try $A=1$, $B=3$: $$ D = 9 - (1 + 3) = 5 $$ Side 2: $B + E + C = 9$ with $B=3$, $C=2$: $$ E = 9 - (3 + 2) = 4 $$ Side 3: $C + F + A = 9$ with $C=2$, $A=1$: $$ F = 9 - (2 + 1) = 6 $$ 13. **Check if $D,E,F = 5,4,6$ are distinct and unused:** Yes, digits 4,5,6 are all distinct and unused. 14. **Solution for sum 9:** $$ A=1, B=3, C=2, D=5, E=4, F=6 $$ --- ### Case (ii): sum = 10 15. **Repeat steps 4-9 with sum 10:** $$ 3 \times 10 = 30 $$ $$ 2(A + B + C) + (D + E + F) = 30 $$ $$ A + B + C + D + E + F = 21 $$ Substitute: $$ 2(A + B + C) + 21 - (A + B + C) = 30 \\ (A + B + C) + 21 = 30 \\ A + B + C = 9 $$ 16. **Then:** $$ D + E + F = 21 - 9 = 12 $$ 17. **Find triples $A,B,C$ with sum 9 from digits 1 to 6:** Possible triples: - (1,2,6) - (1,3,5) - (2,3,4) 18. **Try $A,B,C = (2,3,4)$:** Side 1: $A + D + B = 10$ with $A=2$, $B=3$: $$ D = 10 - (2 + 3) = 5 $$ Side 2: $B + E + C = 10$ with $B=3$, $C=4$: $$ E = 10 - (3 + 4) = 3 $$ $E=3$ conflicts with $B=3$ (already used), so discard. 19. **Try $A,B,C = (1,3,5)$:** Side 1: $A + D + B = 10$ with $A=1$, $B=3$: $$ D = 10 - (1 + 3) = 6 $$ Side 2: $B + E + C = 10$ with $B=3$, $C=5$: $$ E = 10 - (3 + 5) = 2 $$ Side 3: $C + F + A = 10$ with $C=5$, $A=1$: $$ F = 10 - (5 + 1) = 4 $$ 20. **Check if $D,E,F = 6,2,4$ are distinct and unused:** Digits used: $A,B,C = 1,3,5$ and $D,E,F = 6,2,4$ are all distinct and cover 1 to 6. 21. **Solution for sum 10:** $$ A=1, B=3, C=5, D=6, E=2, F=4 $$ --- **Final answers:** - For sum 9: $\boxed{(A,B,C,D,E,F) = (1,3,2,5,4,6)}$ - For sum 10: $\boxed{(A,B,C,D,E,F) = (1,3,5,6,2,4)}