Puzzle for February 15, 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
In eq.5, replace B with A + E (from eq.4): A + C = A + E + E which becomes A + C = A + 2×E Subtract A from each side of the equation above: A + C - A = A + 2×E - A which makes eq.5a) C = 2×E
Hint #2
In eq.2, replace C with 2×E: eq.2a) B = 2×E + D
Hint #3
In eq.6, substitute 2×E + D for B (from eq.2a): E + F = 2×E + D + D which becomes E + F = 2×E + 2×D Subtract E from each side of the above equation: E + F - E = 2×E + 2×D - E which becomes eq.6a) F = E + 2×D
Hint #4
Substitute 2×E for C, and E + 2×D for F (from eq.6a) into eq.3: 2×E = D + E + 2×D which becomes 2×E = 3×D + E Subtract E from both sides of the above equation: 2×E - E = 3×D + E - E which makes E = 3×D
Hint #5
Substitute (3×D) for E in eq.5a: C = 2×(3×D) which makes C = 6×D
Hint #6
Substitute 3×D for E in eq.6a: F = 3×D + 2×D which makes F = 5×D
Hint #7
Substitute 6×D for C in eq.2: B = 6×D + D which makes B = 7×D
Hint #8
Substitute 7×D for B, and 3×D for E in eq.4: 7×D = A + 3×D Subtract 3×D from both sides of the equation above: 7×D - 3×D = A + 3×D - 3×D which makes 4×D = A
Solution
Substitute 4×D for A, 7×D for B, 6×D for C, 3×D for E, and 5×D for F in eq.1: 4×D + 7×D + 6×D + D + 3×D + 5×D = 26 which simplifies to 26×D = 26 Divide both sides of the above equation by 26: 26×D ÷ 26 = 26 ÷ 26 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 7×D = 7 × 1 = 7 C = 6×D = 6 × 1 = 6 E = 3×D = 3 × 1 = 3 F = 5×D = 5 × 1 = 5 and ABCDEF = 476135