Puzzle for May 2, 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
Add C to both sides of eq.3: C - E + C = A - C + C which becomes eq.3a) 2×C - E = A
Hint #2
In eq.5, replace F with C + D (from eq.2), and A with 2×C - E (from eq.3a): E + C + D = 2×C - E + D In the above equation, subtract C and D from both sides, and add E to both sides: E + C + D - C - D + E = 2×C - E + D - C - D + E which simplifies to 2×E = C
Hint #3
In eq.3a, replace C with (2×E): 2×(2×E) - E = A which becomes 4×E - E = A which makes 3×E = A
Hint #4
In eq.4, substitute 2×E for C: D - E = 2×E - D Add E and D to both sides of the above equation: D - E + E + D = 2×E - D + E + D which makes 2×D = 3×E Divide both sides by 2: 2×D ÷ 2 = 3×E ÷ 2 which makes D = 1½×E
Hint #5
Substitute 2×E for C, and 1½×E for D in eq.2: F = 2×E + 1½×E which makes F = 3½×E
Hint #6
Substitute 3×E for A, and 3½×E for F in eq.6: 3×E + E = B + 3½×E - 3×E which becomes 4×E = B + ½×E Subtract ½×E from each side of the equation above: 4×E - ½×E = B + ½×E - ½×E which makes 3½×E = B
Solution
Substitute 3×E for A, 3½×E for B and F, 2×E for C, and 1½×E for D in eq.1: 3×E + 3½×E + 2×E + 1½×E + E + 3½×E = 29 which simplifies to 14½×E = 29 Divide both sides of the above equation by 14½: 14½×E ÷ 14½ = 29 ÷ 14½ which means E = 2 making A = 3×E = 3 × 2 = 6 B = F = 3½×E = 3½ × 2 = 7 C = 2×E = 2 × 2 = 4 D = 1½×E = 1½ × 2 = 3 and ABCDEF = 674327