Puzzle for June 8, 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, replace B + C with A + E (from eq.2): A + D = A + E + E which becomes A + D = A + 2×E Subtract A from each side of the equation above: A + D - A = A + 2×E - A which makes D = 2×E
Hint #2
Add C to both sides of eq.6: C - A - E + C = B - C - D + C which becomes eq.6a) 2×C - A - E = B - D Subtract D from each side of eq.4: A + D - D = B + C + E - D which becomes A = B + C + E - D which may be written as eq.4a) A = B - D + C + E
Hint #3
In eq.4a, replace B - D with 2×C - A - E (from eq.6a): A = 2×C - A - E + C + E which becomes A = 3×C - A Add A to both sides of the above equation: A + A = 3×C - A + A which makes 2×A = 3×C Divide both sides by 2: 2×A ÷ 2 = 3×C ÷ 2 which makes A = 1½×C
Hint #4
In eq.2, substitute 1½×C for A: B + C = 1½×C + E Subtract C from each side of the above equation: B + C - C = 1½×C + E - C which becomes eq.2a) B = ½×C + E
Hint #5
In eq.3, substitute 1½×C for A: E + F = 1½×C + C which becomes E + F = 2½×C Subtract E from both sides of the equation above: E + F - E = 2½×C - E which becomes eq.3a) F = 2½×C - E
Hint #6
Substitute 2½×C - E for F (from eq.3a), 1½×C for A, 2×E for D, and (½×C + E) for B (from eq.2a) in eq.5: 2½×C - E - 1½×C + 2×E = 1½×C - (½×C + E) + E which becomes C + E = 1½×C - ½×C - E + E which becomes C + E = C Subtract C from both sides of the above equation: C + E - C = C - C which makes E = 0 and also makes D = 2×E = 2×0 = 0
Hint #7
Substitute 0 for E in eq.3a: F = 2½×C - 0 which makes F = 2½×C
Hint #8
Substitute 0 for E in eq.2a: B = ½×C + 0 which makes B = ½×C
Solution
Substitute 1½×C for A, ½×C for B, 0 for D and E, and 2½×C for F in eq.1: 1½×C + ½×C + C + 0 + 0 + 2½×C = 11 which simplifies to 5½×C = 11 Divide both sides of the above equation by 5½: 5½×C ÷ 5½ = 11 ÷ 5½ which means C = 2 making A = 1½×C = 1½ × 2 = 3 B = ½×C = ½ × 2 = 1 F = 2½×C = 2½ × 2 = 5 and ABCDEF = 312005