Puzzle for October 17, 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 D to both sides of eq.6: D + E + F + D = A + B + C - D + D which becomes eq.6a) 2×D + E + F = A + B + C Add C to both sides of eq.5: C + E + F + C = A + B - C + C which becomes eq.5a) 2×C + E + F = A + B
Hint #2
In eq.6a, replace A + B with 2×C + E + F (from eq.5a): 2×D + E + F = 2×C + E + F + C which becomes 2×D + E + F = 3×C + E + F Subtract E and F from each side of the equation above: 2×D + E + F - E - F = 3×C + E + F - E - F which simplifies to 2×D = 3×C Divide both sides by 2: 2×D ÷ 2 = 3×C ÷ 2 which makes D = 1½×C
Hint #3
In eq.2, substitute 1½×C for D: B = C + 1½×C which makes B = 2½×C
Hint #4
Substitute 2½×C for B in eq.3: E = 2½×C + C which makes E = 3½×C
Hint #5
Substitute 1½×C for D, 2½×C for B, and 3½×C for E in eq.4: A + 1½×C = 2½×C + 3½×C which becomes A + 1½×C = 6×C Subtract 1½×C from both sides of the above equation: A + 1½×C - 1½×C = 6×C - 1½×C which makes A = 4½×C
Hint #6
Substitute 3½×C for E, 4½×C for A, and 2½×C for B in eq.5a: 2×C + 3½×C + F = 4½×C + 2½×C which becomes 5½×C + F = 7×C Subtract 5½×C from each side of the above equation: 5½×C + F - 5½×C = 7×C - 5½×C which makes F = 1½×C
Solution
Substitute 4½×C for A, 2½×C for B, 1½×C for D and F, and 3½×C for E in eq.1: 4½×C + 2½×C + C + 1½×C + 3½×C + 1½×C = 29 which simplifies to 14½×C = 29 Divide both sides of the above equation by 14½: 14½×C ÷ 14½ = 29 ÷ 14½ which means C = 2 making A = 4½×C = 4½ × 2 = 9 B = 2½×C = 2½ × 2 = 5 D = F = 1½×C = 1½ × 2 = 3 E = 3½×C = 3½ × 2 = 7 and ABCDEF = 952373