Puzzle for August 2, 2021 ( )
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.6, replace A with B + D (from eq.4): B + C = B + D + D which becomes B + C = B + 2×D Subtract B from each side of the above equation: B + C – B = B + 2×D – B which makes C = 2×D
Hint #2
In eq.3, replace C with 2×D: B = 2×D + D which makes B = 3×D
Hint #3
In eq.6, substitute 3×D for B, and 2×D for C: 3×D + 2×D = A + D which becomes 5×D = A + D Subtract D from each side of the equation above: 5×D – D = A + D – D which makes 4×D = A
Hint #4
Substitute 2×D for C in eq.5: E – D = 2×D – E Add D and E to both sides of the above equation: E – D + D + E = 2×D – E + D + E which makes 2×E = 3×D Divide both sides by 2: 2×E ÷ 2 = 3×D ÷ 2 which makes E = 1½×D
Hint #5
Substitute 3×D for B, and 1½×D for E in eq.2: F = 3×D + 1½×D which makes F = 4½×D
Solution
Substitute 4×D for A, 3×D for B, 2×D for C, 1½×D for E, and 4½×D for F in eq.1: 4×D + 3×D + 2×D + D + 1½×D + 4½×D = 32 which simplifies to 16×D = 32 Divide both sides of the above equation by 16: 16×D ÷ 16 = 32 ÷ 16 which means D = 2 making A = 4×D = 4 × 2 = 8 B = 3×D = 3 × 2 = 6 C = 2×D = 2 × 2 = 4 E = 1½×D = 1½ × 2 = 3 F = 4½×D = 4½ × 2 = 9 and ABCDEF = 864239