Puzzle for March 25, 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
eq.6 may be written as: B + C + E = A + F + D In the above equation, replace A + F with B + C + D (from eq.3): B + C + E = B + C + D + D which becomes B + C + E = B + C + 2×D Subtract B and C from each side of the above equation: B + C + E - B - C = B + C + 2×D - B - C which simplifies to E = 2×D
Hint #2
In eq.5, replace E with 2×D: D + 2×D + F = A + C - D which becomes 3×D + F = A + C - D Add D to both sides of the equation above: 3×D + F + D = A + C - D + D which becomes eq.5a) 4×D + F = A + C
Hint #3
In eq.4, substitute 2×D for E: 2×D + F = A + B - 2×D Add 2×D to both sides of the above equation: 2×D + F + 2×D = A + B - 2×D + 2×D which becomes eq.4a) 4×D + F = A + B
Hint #4
Substitute A + B for 4×D + F (from eq.4a) into eq.5a: A + B = A + C Subtract A from each side of the equation above: A + B - A = A + C - A which makes B = C
Hint #5
eq.5 may be written as: D + F + E = A + C - D Substitute A + E for D + F (from eq.2) in the above equation: A + E + E = A + C - D which becomes A + 2×E = A + C - D Subtract A from each side: A + 2×E - A = A + C - D - A which becomes eq.5b) 2×E = C - D
Hint #6
Substitute (2×D) for E in eq.5b: 2×(2×D) = C - D which becomes 4×D = C - D Add D to both sides of the equation above: 4×D + D = C - D + D which makes 5×D = C and also makes 5×D = C = B
Hint #7
Substitute 5×D for B and C in eq.3: A + F = 5×D + 5×D + D which makes eq.3a) A + F = 11×D
Hint #8
Substitute 2×D for E in eq.2: D + F = A + 2×D Subtract D from each side of the above equation: D + F - D = A + 2×D - D which becomes eq.2a) F = A + D
Hint #9
Substitute A + D for F (from eq.2a) into eq.3a: A + A + D = 11×D which becomes 2×A = 10×D Divide both sides of the above equation by 2: 2×A ÷ 2 = 10×D ÷ 2 which makes A = 5×D
Hint #10
Substitute 5×D for A in eq.2a: F = 5×D + D which makes F = 6×D
Solution
Substitute 5×D for A and B and C, 2×D for E, and 6×D for F in eq.1: 5×D + 5×D + 5×D + D + 2×D + 6×D = 24 which simplifies to 24×D = 24 Divide both sides of the above equation by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = B = C = 5×D = 5 × 1 = 5 E = 2×D = 2 × 1 = 2 F = 6×D = 6 × 1 = 6 and ABCDEF = 555126