Puzzle for May 23, 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.6, replace A with C + F (from eq.2): F – C = C + F – F which becomes F – C = C Add C to both sides of the above equation: F – C + C = C + C which makes F = 2×C
Hint #2
In eq.2, replace F with 2×C: A = C + 2×C which makes A = 3×C
Hint #3
In eq.5, substitute 2×C for F, and 3×C for A: E – 2×C = 3×C + C which becomes E – 2×C = 4×C Add 2×C to both sides of the equation above: E – 2×C + 2×C = 4×C + 2×C which makes E = 6×C
Hint #4
Substitute 6×C for E, and 2×C for F in eq.3: B = 6×C + 2×C which makes B = 8×C
Hint #5
Substitute 8×C for B in eq.4: D = 8×C + C which makes D = 9×C
Solution
Substitute 3×C for A, 8×C for B, 9×C for D, 6×C for E, and 2×C for F in eq.1: 3×C + 8×C + C + 9×C + 6×C + 2×C = 29 which simplifies to 29×C = 29 Divide both sides of the above equation by 29: 29×C ÷ 29 = 29 ÷ 29 which means C = 1 making A = 3×C = 3 × 1 = 3 B = 8×C = 8 × 1 = 8 D = 9×C = 9 × 1 = 9 E = 6×C = 6 × 1 = 6 F = 2×C = 2 × 1 = 2 and ABCDEF = 381962