Puzzle for December 1, 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.4, add A and F to both sides, and subtract B from both sides: B - A + A + F - B = A - F + A + F - B which becomes eq.4a) F = 2×A - B
Hint #2
In eq.6, replace F with 2×A - B (from eq.4a): C + 2×A - B = A + B - C In the above equation, subtract A from both sides, and add B and C to both sides: C + 2×A - B - A + B + C = A + B - C - A + B + C which becomes eq.6a) A + 2×C = 2×B
Hint #3
In eq.6a, substitute (C + E) for B (from eq.2): A + 2×C = 2×(C + E) which becomes A + 2×C = 2×C + 2×E Subtract 2×C from each side of the equation above: A + 2×C - 2×C = 2×C + 2×E - 2×C which makes A = 2×E
Hint #4
Substitute (2×E) for A, and (C + E) for B (from eq.2) into eq.4a: F = 2×(2×E) - (C + E) which becomes F = 4×E - C - E which becomes eq.4b) F = 3×E - C
Hint #5
Substitute (3×E - C) for F (from eq.4b) in eq.3: D - (3×E - C) = (3×E - C) - C which becomes D - 3×E + C = 3×E - 2×C In the equation above, add 3×E to both sides, and subtract C from both sides: D - 3×E + C + 3×E - C = 3×E - 2×C + 3×E - C which becomes eq.3a) D = 6×E - 3×C
Hint #6
Substitute 2×E for A, and 6×E - 3×C for D (from eq.3a) in eq.5: 2×E + C = 6×E - 3×C + E which becomes 2×E + C = 7×E - 3×C In the equation above, add 3×C to both sides, and subtract 2×E from both sides: 2×E + C + 3×C - 2×E = 7×E - 3×C + 3×C - 2×E which makes 4×C = 5×E Divide both sides by 4: 4×C ÷ 4 = 5×E ÷ 4 which makes C = 1¼×E
Hint #7
Substitute (1¼×E) for C in eq.3a: D = 6×E - 3×(1¼×E) which becomes D = 6×E - 3¾×E which makes D = 2¼×E
Hint #8
Substitute 1¼×E for C in eq.4b: F = 3×E - 1¼×E which makes F = 1¾×E
Hint #9
Substitute 1¼×E for C in eq.2: B = 1¼×E + E which makes B = 2¼×E
Solution
Substitute 2×E for A, 2¼×E for B and D, 1¼×E for C, and 1¾×E for F in eq.1: 2×E + 2¼×E + 1¼×E + 2¼×E + E + 1¾×E = 42 which simplifies to 10½×E = 42 Divide both sides of the above equation by 10½: 10½×E ÷ 10½ = 42 ÷ 10½ which means E = 4 making A = 2×E = 2 × 4 = 8 B = D = 2¼×E = 2¼ × 4 = 9 C = 1¼×E = 1¼ × 4 = 5 F = 1¾×E = 1¾ × 4 = 7 and ABCDEF = 895947