Puzzle for October 21, 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 positive integer.
Scratchpad
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Hint #1
In eq.4, replace B with A + C (from eq.2): C + D = A + C + F Subtract C from each side of the equation above: C + D - C = A + C + F - C which becomes eq.4a) D = A + F
Hint #2
In eq.3, replace D with A + F (from eq.4a): A + F = E + F Subtract F from both sides of the equation above: A + F - F = E + F - F which becomes A = E
Hint #3
In eq.2, substitute E for A: eq.2a) B = E + C
Hint #4
Substitute E + C for B (from eq.2a) in eq.5: E + C - C = C - D which becomes E = C - D In the above equation, subtract E from both sides, and add D to both sides: E - E + D = C - D - E + D which becomes eq.5a) D = C - E
Hint #5
Substitute C - E for D (from eq.5a) into eq.3: C - E = E + F Subtract E from each side of the above equation: C - E - E = E + F - E which becomes eq.3a) C - 2×E = F
Hint #6
Substitute (C - E) for D (from eq.5a), (C - 2×E) for F (from eq.3a), E for A, and E + C for B (from eq.2a) in eq.6: (C - E) × (C - 2×E) = (E + E) × (C + E + C) which becomes C² - 2×E×C - E×C + 2×E² = 2×E × (2×C + E) which becomes C² - 3×E×C + 2×E² = 4×E×C + 2×E² In the above equation, add 3×E×C to both sides, and subtract 2×E² from both sides: C² - 3×E×C + 2×E² + 3×E×C - 2×E² = 4×E×C + 2×E² + 3×E×C - 2×E² which simplifies to C² = 7×E×C Divide both sides by C: C² ÷ C = 7×E×C ÷ C which makes C = 7×E
Hint #7
Substitute 7×E for C in eq.3a: 7×E - 2×E = F which makes 5×E = F
Hint #8
Substitute 7×E for C in eq.5a: D = 7×E - E which makes D = 6×E
Hint #9
Substitute 7×E for C in eq.2a: B = E + 7×E which makes B = 8×E
Solution
Substitute E for A, 8×E for B, 7×E for C, 6×E for D, and 5×E for F in eq.1: E + 8×E + 7×E + 6×E + E + 5×E = 28 which simplifies to 28×E = 28 Divide both sides of the above equation by 28: 28×E ÷ 28 = 28 ÷ 28 which means E = 1 making A = E = 1 B = 8×E = 8 × 1 = 8 C = 7×E = 7 × 1 = 7 D = 6×E = 6 × 1 = 6 F = 5×E = 5 × 1 = 5 and ABCDEF = 187615