Puzzle for September 24, 2019 ( )
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.2, replace E with B + C + D (from eq.5): A + B + D = C + B + C + D which becomes A + B + D = 2×C + B + D Subtract B and D from both sides of the above equation: A + B + D – B – D = 2×C + B + D – B – D which simplifies to eq.2a) A = 2×C
Hint #2
In eq.6, replace B with C + D (from eq.3), and replace A with 2×C (from eq.2a): C + D + D = 2×C – D In the above equation, add D to both sides, and subtract C from each side: C + D + D + D – C = 2×C – D + D – C which means 3×D = C
Hint #3
In eq.2a, replace C with (3×D): A = 2×(3×D) which makes A = 6×D
Hint #4
In eq.4, substitute 6×D for A: D + F = 6×D Subtract D from both sides of the above equation: D + F – D = 6×D – D which makes F = 5×D
Hint #5
Substitute 6×D for A in eq.6: B + D = 6×D – D Subtract D from both sides of the above equation: B + D – D = 6×D – D – D which makes B = 4×D
Hint #6
Substitute 4×D for B, and 3×D for C in eq.5: E = 4×D + 3×D + D which makes E = 8×D
Solution
Substitute 6×D for A, 4×D for B, 3×D for C, 8×D for E, and 5×D for F in eq.1: 6×D + 4×D + 3×D + D + 8×D + 5×D = 27 which simplifies to 27×D = 27 Divide both sides of the equation above by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = 6×D = 6 × 1 = 6 B = 4×D = 4 × 1 = 4 C = 3×D = 3 × 1 = 3 E = 8×D = 8 × 1 = 8 F = 5×D = 5 × 1 = 5 and ABCDEF = 643185