Puzzle for February 3, 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.4, replace E + F with A + C (from eq.5): B + C = A + A + C which becomes B + C = 2×A + C Subtract C from each side of the equation above: B + C – C = 2×A + C – C which makes B = 2×A
Hint #2
In eq.2, replace B with 2×A: E = A + 2×A which makes E = 3×A
Hint #3
In eq.3, substitute 3×A for E: C + D + F = A + 3×A which becomes eq.3a) C + D + F = 4×A
Hint #4
eq.1 may be written as: A + B + E + C + D + F = 30 Substitute 2×A for B, 3×A for E, and 4×A for C + D + F (from eq.3a) in the above equation: A + 2×A + 3×A + 4×A = 30 which makes 10×A = 30 Divide both sides by 10: 10×A ÷ 10 = 30 ÷ 10 which means A = 3 making B = 2×A = 2 × 3 = 6 E = 3×A = 3 × 3 = 9
Hint #5
Substitute 3 for A, and 9 for E in eq.5: 9 + F = 3 + C Subtract 9 from each side of the above equation: 9 + F – 9 = 3 + C – 9 which becomes eq.5a) F = C – 6
Hint #6
Substitute 3 for A in eq.6: C = 3 + D Subtract 3 from each side of the equation above: C – 3 = 3 + D – 3 which becomes eq.6a) C – 3 = D
Solution
Substitute C – 3 for D (from eq.6a), C – 6 for F (from eq.5a), 3 for A, and 9 for E in eq.3: C + C – 3 + C – 6 = 3 + 9 which becomes 3×C – 9 = 12 Add 9 to both sides of the equation above: 3×C – 9 + 9 = 12 + 9 which makes 3×C = 21 Divide both sides of the equation above by 3: 3×C ÷ 3 = 21 ÷ 3 which means C = 7 making D = C – 3 = 7 – 3 = 4 (from eq.6a) F = C – 6 = 7 – 6 = 1 (from eq.5a) and ABCDEF = 367491