Puzzle for November 4, 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 + E (from eq.3): C + F = C + E + E - C which becomes eq.6a) C + F = 2×E
Hint #2
In eq.4, replace F with D + E (from eq.2): B + D = E + D + E - D which becomes eq.4a) B + D = 2×E
Hint #3
In eq.4a, substitute C + F for 2×E (from eq.6a): B + D = C + F which may be written as eq.4b) C + F = B + D Add D to both sides of eq.5: D + F + D = B + C - D + D which becomes eq.5a) 2×D + F = B + C
Hint #4
Subtract the left and right sides of eq.4b from the left and right sides of eq.5a, respectively: 2×D + F - (C + F) = B + C - (B + D) which becomes 2×D + F - C - F = B + C - B - D which becomes 2×D - C = C - D Add C and D to both sides of the above equation: 2×D - C + C + D = C - D + C + D which makes 3×D = 2×C Divide both sides by 2: 3×D ÷ 2 = 2×C ÷ 2 which makes 1½×D = C
Hint #5
Substitute 1½×D for C in eq.4b: 1½×D + F = B + D Subtract D from each side of the equation above: 1½×D + F - D = B + D - D which becomes eq.4c) ½×D + F = B
Hint #6
Substitute ½×D + F for B (from eq.4c) into eq.4: ½×D + F + D = E + F - D which becomes 1½×D + F = E + F - D In the above equation, subtract F from both sides, and add D to both sides: 1½×D + F - F + D = E + F - D - F + D which simplifies to 2½×D = E
Hint #7
Substitute 1½×D for C, and 2½×D for E in eq.3: A = 1½×D + 2½×D which makes A = 4×D
Hint #8
Substitute 2½×D for E in eq.2: F = D + 2½×D which makes F = 3½×D
Hint #9
Substitute 3½×D for F in eq.4c: ½×D + 3½×D = B which makes 4×D = B
Solution
Substitute 4×D for A and B, 1½×D for C, 2½×D for E, and 3½×D for F in eq.1: 4×D + 4×D + 1½×D + D + 2½×D + 3½×D = 33 which simplifies to 16½×D = 33 Divide both sides of the above equation by 16½: 16½×D ÷ 16½ = 33 ÷ 16½ which means D = 2 making A = B = 4×D = 4 × 2 = 8 C = 1½×D = 1½ × 2 = 3 E = 2½×D = 2½ × 2 = 5 F = 3½×D = 3½ × 2 = 7 and ABCDEF = 883257