Puzzle for October 6, 2020 ( )
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 F with B + D (from eq.4): C = B + B + D which becomes eq.6a) C = 2×B + D
Hint #2
In eq.2, replace C with 2×B + D (from eq.6a): 2×B + D = D + E Subtract D from both sides of the above equation: 2×B + D – D = D + E – D which becomes 2×B = E
Hint #3
In eq.5, substitute 2×B for E: B + 2×B = A which makes 3×B = A
Hint #4
Substitute 2×B for E, and 3×B for A in eq.3: D – 2×B = 3×B – D Add 2×B and D to both sides of the equation above: D – 2×B + 2×B + D = 3×B – D + 2×B + D which becomes 2×D = 5×B Divide both sides by 2: 2×D ÷ 2 = 5×B ÷ 2 which makes D = 2½×B
Hint #5
Substitute 2½×B for D in eq.4: F = B + 2½×B which makes F = 3½×B
Hint #6
Substitute 3½×B for F in eq.6: C = B + 3½×B which makes C = 4½×B
Solution
Substitute 3×B for A, 4½×B for C, 2½×B for D, 2×B for E, and 3½×B for F in eq.1: 3×B + B + 4½×B + 2½×B + 2×B + 3½×B = 33 which simplifies to 16½×B = 33 Divide both sides of the equation above by 16½: 16½×B ÷ 16½ = 33 ÷ 16½ which means B = 2 making A = 3×B = 3 × 2 = 6 C = 4½×B = 4½ × 2 = 9 D = 2½×B = 2½ × 2 = 5 E = 2×B = 2 × 2 = 4 F = 3½×B = 3½ × 2 = 7 and ABCDEF = 629547