Puzzle for February 15, 2019 ( )
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.
* CD, DE, and EF are 2-digit numbers (not C×D, D×E, or E×F).
Scratchpad
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Hint #1
In eq.3, replace E with C + F (from eq.5): C + F = A + D + F Subtract both A and F from each side of the above equation: C + F - A - F = A + D + F - A - F which becomes eq.3a) C - A = D
Hint #2
In eq.2, replace D with C - A (from eq.3a): C = A + B + C - A which becomes C = B + C Subtract C from each side of the equation above: C - C = B + C - C which means 0 = B
Hint #3
In eq.4, substitute A + B + D for C (from eq.2): D + F = A + B + A + B + D which is the same as D + F = 2×A + 2×B + D Substitute 0 for B, and subtract D from each side of the equation above: D + F - D = 2×A + 2×0 + D - D which simplifies to 2×A = F
Hint #4
In eq.4, substitute 0 for B: D + F = A + 0 + C which becomes D + F = A + C Substitute A + C for D + F in eq.3: E = A + A + C which becomes eq.3b) E = 2×A + C
Hint #5
eq.6 may be written as: 10×E + F - (10×C + D) = A + B + 10×D + E which is the same as 10×E + F - 10×C - D = A + B + 10×D + E Add (D - E) to both sides of the above equation: 10×E + F - 10×C - D + (D - E) = A + B + 10×D + E + (D - E) which becomes eq.6a) 9×E + F - 10×C = A + B + 11×D
Hint #6
Substitute (2×A + C) for E (from eq.3b), 2×A for F, 0 for B, and (C - A) for D (from eq.3a) in eq.6a: 9×(2×A + C) + 2×A - 10×C = A + 0 + 11×(C - A) which is equivalent to 18×A + 9×C + 2×A - 10×C = A + 0 + 11×C - 11×A which simplifies to 20×A - C = 11×C - 10×A Add 10×A + C to both sides of the above equation: 20×A - C + 10×A + C = 11×C - 10×A + 10×A + C which makes 30×A = 12×C Divide each side by 12: 30×A ÷ 12 = 12×C ÷ 12 which means 2½×A = C
Hint #7
Substitute 2½×A for C in eq.3a: 2½×A - A = D which makes 1½×A = D
Hint #8
Substitute 2½×A for C in eq.3b: E = 2½×A + 2×A which makes E = 4½×A
Solution
Substitute 0 for B, 2½×A for C, 1½×A for D, 4½×A for E, and 2×A for F in eq.1: A + 0 + 2½×A + 1½×A + 4½×A + 2×A = 23 which simplifies to 11½×A = 23 Divide each side of the above equation by 11½: 11½×A ÷ 11½ = 23 ÷ 11½ which means A = 2 making C = 2½×A = 2½ × 2 = 5 D = 1½×A = 1½ × 2 = 3 E = 4½×A = 4½ × 2 = 9 F = 2×A = 2 × 2 = 4 and ABCDEF = 205394