Puzzle for November 11, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add F to both sides of eq.6: A + C - F + F = D + E + F + F which becomes eq.6a) A + C = D + E + 2×F Add F and C to both sides of eq.4: D - F + F + C = F - C + F + C which becomes eq.4a) D + C = 2×F Add F and A to both sides of eq.5: E - F + F + A = F - A + F + A which becomes eq.5a) E + A = 2×F
Hint #2
In eq.6a, replace 2×F with D + C (from eq.4a): A + C = D + E + D + C which becomes A + C = 2×D + E + C Subtract C from each side of the equation above: A + C - C = 2×D + E + C - C which becomes eq.6b) A = 2×D + E
Hint #3
In eq.6a, replace 2×F with E + A (from eq.5a): A + C = D + E + E + A which becomes A + C = D + 2×E + A Subtract A from each side of the equation above: A + C - A = D + 2×E + A - A which becomes eq.6c) C = D + 2×E
Hint #4
In eq.6a, substitute 2×D + E for A (from eq.6b) and D + 2×E for C (from eq.6c): 2×D + E + D + 2×E = D + E + 2×F which becomes 3×D + 3×E = D + E + 2×F Subtract D and E from both sides of the above equation: 3×D + 3×E - D - E = D + E + 2×F - D - E which becomes 2×D + 2×E = 2×F Divide both sides by 2: (2×D + 2×E) ÷ 2 = 2×F ÷ 2 which becomes eq.6d) D + E = F
Hint #5
Substitute D + E for F (from eq.6d), and 2×D + E for A (from eq.6b) in eq.2: B + D + E = 2×D + E + D which becomes B + D + E = 3×D + E Subtract D and E from both sides of the above equation: B + D + E - D - E = 3×D + E - D - E which simplifies to B = 2×D Substitute 2×D + E for A (from eq.6b), and 2×D for B in eq.3: C + E = 2×D + E + 2×D which becomes C + E = 4×D + E Subtract E from both sides of the equation above: C + E - E = 4×D + E - E which makes C = 4×D
Hint #6
Substitute 4×D for C in eq.4a: D + 4×D = 2×F which makes 5×D = 2×F Divide both of the above equation sides by 2: 5×D ÷ 2 = 2×F ÷ 2 which makes 2½×D = F
Hint #7
Substitute 2½×D for F in eq.6d: D + E = 2½×D Subtract D from each side of the equation above: D + E - D = 2½×D - D which makes E = 1½×D
Hint #8
Substitute 1½×D for E in eq.6b: A = 2×D + 1½×D which makes A = 3½×D
Solution
Substitute 3½×D for A, 2×D for B, 4×D for C, 1½×D for E, and 2½×D for F in eq.1: 3½×D + 2×D + 4×D + D + 1½×D + 2½×D = 29 which simplifies to 14½×D = 29 Divide both sides of the above equation by 14½: 14½×D ÷ 14½ = 29 ÷ 14½ which means D = 2 making A = 3½×D = 3½ × 2 = 7 B = 2×D = 2 × 2 = 4 C = 4×D = 4 × 2 = 8 E = 1½×D = 1½ × 2 = 3 F = 2½×D = 2½ × 2 = 5 and ABCDEF = 748235