Puzzle for February 2, 2022 ( )
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.3, replace E with C + D (from eq.2): D + C + D = B + C which becomes 2×D + C = B + C Subtract C from each side of the above equation: 2×D + C – C = B + C – C which makes 2×D = B
Hint #2
In eq.5, replace E with C + D (from eq.2): A + D = C + C + D which becomes A + D = 2×C + D Subtract D from each side of the above equation: A + D – D = 2×C + D – D which makes eq.5a) A = 2×C
Hint #3
In eq.6, substitute C + D for E (from eq.2), 2×C for A (from eq.5a), and 2×D for B: C + D + F = 2×C + 2×D + C which becomes C + D + F = 3×C + 2×D Subtract C and D from both sides of the above equation: C + D + F – C – D = 3×C + 2×D – C – D which becomes eq.6a) F = 2×C + D
Hint #4
Substitute 2×D for B, 2×C + D for F (from eq.6a), and 2×C for A (from eq.5a) in eq.4: 2×D + 2×C + D = 2×C + C which becomes 3×D + 2×C = 3×C Subtract 2×C from each side of the equation above: 3×D + 2×C – 2×C = 3×C – 2×C which makes 3×D = C
Hint #5
Substitute (3×D) for C in eq.5a: A = 2×(3×D) which makes A = 6×D
Hint #6
Substitute 3×D for C in eq.2: 3×D + D = E which makes 4×D = E
Hint #7
Substitute (3×D) for C in eq.6a: F = 2×(3×D) + D which becomes F = 6×D + D which makes F = 7×D
Solution
Substitute 6×D for A, 2×D for B, 3×D for C, 4×D for E, and 7×D for F in eq.1: 6×D + 2×D + 3×D + D + 4×D + 7×D = 23 which simplifies to 23×D = 23 Divide both sides of the above equation by 23: 23×D ÷ 23 = 23 ÷ 23 which means D = 1 making A = 6×D = 6 × 1 = 6 B = 2×D = 2 × 1 = 2 C = 3×D = 3 × 1 = 3 E = 4×D = 4 × 1 = 4 F = 7×D = 7 × 1 = 7 and ABCDEF = 623147