Puzzle for February 9, 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.4, replace F with C + D (from eq.2): E + C + D = A + D Subtract D from each side of the equation above: E + C + D - D = A + D - D which becomes E + C = A which may be written as eq.4a) C + E = A
Hint #2
In eq.5, replace D with A + B + C (from eq.3): A + B + C + E = B + F Subtract B from both sides of the equation above: A + B + C + E - B = B + F - B which becomes eq.5a) A + C + E = F
Hint #3
In eq.5a, substitute A for C + E (from eq.4a): A + A = F which makes eq.5b) 2×A = F
Hint #4
In eq.6, substitute C + E for A (from eq.4a): B + E = C + E + C which becomes B + E = 2×C + E Subtract E from both sides of the above equation: B + E - E = 2×C + E - E which makes B = 2×C
Hint #5
Substitute 2×C for B, and C + D for F (from eq.2) in eq.5: D + E = 2×C + C + D which becomes D + E = 3×C + D Subtract D from each side of the equation above: D + E - D = 3×C + D - D which makes E = 3×C
Hint #6
Substitute 3×C for E in eq.4a: C + 3×C = A which makes 4×C = A
Hint #7
Substitute (4×C) for A in eq.5b: 2×(4×C) = F which makes 8×C = F
Hint #8
Substitute 8×C for F in eq.2: 8×C = C + D Subtract C from each side of the equation above: 8×C - C = C + D - C which makes 7×C = D
Solution
Substitute 4×C for A, 2×C for B, 7×C for D, 3×C for E, and 8×C for F in eq.1: 4×C + 2×C + C + 7×C + 3×C + 8×C = 25 which simplifies to 25×C = 25 Divide both sides of the above equation by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = 4×C = 4 × 1 = 4 B = 2×C = 2 × 1 = 2 D = 7×C = 7 × 1 = 7 E = 3×C = 3 × 1 = 3 F = 8×C = 8 × 1 = 8 and ABCDEF = 421738