Puzzle for October 11, 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 C + F with B - C + E (from eq.4): B + D + E = A + B - C + E Subtract B and E from each side of the above equation: B + D + E - B - E = A + B - C + E - B - E which simplifies to eq.5a) D = A - C
Hint #2
In eq.2, replace D with A - C (from eq.5a): A = A - C + F In the equation above, subtract A from both sides, and add C to both sides: A - A + C = A - C + F - A + C which makes C = F
Hint #3
In eq.3, substitute C for F: B + C = C + E Subtract C from each side of the equation above: B + C - C = C + E - C which makes B = E
Hint #4
Substitute C for F, and B for E in eq.4: C + C = B - C + B which becomes 2×C = 2×B - C Add C to both sides of the above equation: 2×C + C = 2×B - C + C which makes 3×C = 2×B Divide both sides by 2: 3×C ÷ 2 = 2×B ÷ 2 which makes 1½×C = B and also makes 1½×C = B = E
Hint #5
Substitute 1½×C for B and E, A - C for D (from eq.5a), and C for F in eq.6: A - 1½×C + A - C - 1½×C = 1½×C + C + 1½×C + C which becomes 2×A - 4×C = 5×C Add 4×C to both sides of the above equation: 2×A - 4×C + 4×C = 5×C + 4×C which makes 2×A = 9×C Divide both sides by 2: 2×A ÷ 2 = 9×C ÷ 2 which makes A = 4½×C
Hint #6
Substitute 4½×C for A in eq.5a: D = 4½×C - C which makes D = 3½×C
Solution
Substitute 4½×C for A, 1½×C for B and E, 3½×C for D, and C for F in eq.1: 4½×C + 1½×C + C + 3½×C + 1½×C + C = 26 which simplifies to 13×C = 26 Divide both sides of the above equation by 13: 13×C ÷ 13 = 26 ÷ 13 which means C = 2 making A = 4½×C = 4½ × 2 = 9 B = E = 1½×C = 1½ × 2 = 3 D = 3½×C = 3½ × 2 = 7 F = C = 2 and ABCDEF = 932732