Puzzle for July 11, 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 D + F with A (from eq.3): A + C = B + A Subtract A from each side of the above equation: A + C – A = B + A – A which makes C = B
Hint #2
In eq.4, replace C with B: B + D = B + F Subtract B from each side of the equation above: B + D – B = B + F – B which makes D = F
Hint #3
In eq.3, substitute D for F: A = D + D which makes A = 2×D
Hint #4
Substitute 2×D for A in eq.2: E = 2×D + D which makes E = 3×D
Hint #5
Substitute B for C, 3×D for E, and D for F in eq.5: B + B = D + 3×D + D which makes 2×B = 5×D Divide both sides of the above equation by 2: 2×B ÷ 2 = 5×D ÷ 2 which makes B = 2½×D and also makes C = B = 2½×D
Solution
Substitute 2×D for A, 2½×D for B and C, 3×D for E, and D for F in eq.1: 2×D + 2½×D + 2½×D + D + 3×D + D = 24 which simplifies to 12×D = 24 Divide both sides of the above equation by 12: 12×D ÷ 12 = 24 ÷ 12 which means D = 2 making A = 2×D = 2 × 2 = 4 B = C = 2½×D = 2½ × 2 = 5 E = 3×D = 3 × 2 = 6 F = D = 2 and ABCDEF = 455262