Puzzle for November 2, 2021 ( )
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.2, replace D with B + E (from eq.3), and A with E + F (from eq.5): B + B + E = E + F which becomes 2×B + E = E + F Subtract E from both sides of the above equation: 2×B + E – E = E + F – E which makes 2×B = F
Hint #2
In eq.4, replace F with 2×B: B + 2×B = C which makes 3×B = C
Hint #3
In eq.6, substitute B + D for A (from eq.2), 3×B for C, and 2×B for F: B + D + D = B + 3×B + 2×B which becomes B + 2×D = 6×B Subtract B from both sides of the equation above: B + 2×D – B = 6×B – B which makes 2×D = 5×B Divide both sides by 2: 2×D ÷ 2 = 5×B ÷ 2 which makes D = 2½×B
Hint #4
Substitute 2½×B for D in eq.2: B + 2½×B = A which makes 3½×B = A
Hint #5
Substitute 2½×B for D in eq.3: B + E = 2½×B Subtract B from each side of the equation above: B + E – B = 2½×B – B which makes E = 1½×B
Solution
Substitute 3½×B for A, 3×B for C, 2½×B for D, 1½×B for E, and 2×B for F in eq.1: 3½×B + B + 3×B + 2½×B + 1½×B + 2×B = 27 which simplifies to 13½×B = 27 Divide both sides of the above equation by 13½: 13½×B ÷ 13½ = 27 ÷ 13½ which means B = 2 making A = 3½×B = 3½ × 2 = 7 C = 3×B = 3 × 2 = 6 D = 2½×B = 2½ × 2 = 5 E = 1½×B = 1½ × 2 = 3 F = 2×B = 2 × 2 = 4 and ABCDEF = 726534