Puzzle for August 15, 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.5, replace B wih C + D (from eq.2): C + F = C + D + D which becomes C + F = C + 2×D Subtract C from each side of the equation above: C + F - C = C + 2×D - C which makes F = 2×D
Hint #2
In eq.6, replace A with B + E (from eq.3): E + F = B + E + C - F In the above equation, subtract E from both sides, and add F to both sides: E + F - E + F = B + E + C - F - E + F which becomes eq.6a) 2×F = B + C
Hint #3
In eq.6a, substitute (2×D) for F, and C + D for B (from eq.2): 2×(2×D) = C + D + C which becomes 4×D = 2×C + D Subtract D from each side of the above equation: 4×D - D = 2×C + D - D which makes 3×D = 2×C Divide both sides by 2: 3×D ÷ 2 = 2×C ÷ 2 which makes 1½×D = C
Hint #4
Substitute 1½×D for C in eq.2: B = 1½×D + D which makes B = 2½×D
Hint #5
Substitute 2×D for F, and 1½×D for C in eq.4: D + 2×D = 1½×D + E which becomes 3×D = 1½×D + E Subtract 1½×D from both sides of the equation above: 3×D - 1½×D = 1½×D + E - 1½×D which makes 1½×D = E
Hint #6
Substitute 2½×D for B, and 1½×D for E in eq.3: A = 2½×D + 1½×D which becomes A = 4×D
Solution
Substitute 4×D for A, 2½×D for B, 1½×D for C and E, and 2×D for F in eq.1: 4×D + 2½×D + 1½×D + D + 1½×D + 2×D = 25 which simplifies to 12½×D = 25 Divide both sides of the above equation by 12½: 12½×D ÷ 12½ = 25 ÷ 12½ which means D = 2 making A = 4×D = 4 × 2 = 8 B = 2½×D = 2½ × 2 = 5 C = E = 1½×D = 1½ × 2 = 3 F = 2×D = 2 × 2 = 4 and ABCDEF = 853234