Puzzle for February 12, 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.2, replace E with A + C (from eq.3): B + C = A + C + F Subtract C from both sides of the above equation: B + C – C = A + C + F – C which becomes eq.2a) B = A + F
Hint #2
In eq.6, replace B with A + F (from eq.2a): A + F + D + F = A + E which becomes A + D + 2×F = A + E Subtract A from each side of the equation above: A + D + 2×F – A = A + E – A which becomes eq.6a) D + 2×F = E
Hint #3
In eq.2, substitute C + D for B (from eq.4), and D + 2×F for E (from eq.6a): C + D + C = D + 2×F + F which becomes 2×C + D = D + 3×F Subtract D from each side of the equation above: 2×C + D – D = D + 3×F – D which makes 2×C = 3×F Divide both sides by 2: 2×C ÷ 2 = 3×F ÷ 2 which makes eq.2b) C = 1½×F
Hint #4
Substitute 1½×F for C in eq.2: B + 1½×F = E + F Subtract F from each side of the above equation: B + 1½×F – F = E + F – F which becomes eq.2c) B + ½×F = E
Hint #5
Substitute B + ½×F for E (from eq.2c) in eq.6: B + D + F = A + B + ½×F Subtract B and ½×F from both sides of the equation above: B + D + F – B – ½×F = A + B + ½×F – B – ½×F which becomes eq.6b) D + ½×F = A
Hint #6
Substitute D + ½×F for A (from eq.6b) in eq.5: F = D + ½×F + D which becomes F = 2×D + ½×F Subtract ½×F from both sides of the equation above: F – ½×F = 2×D + ½×F – ½×F which makes ½×F = 2×D Multiply both sides by 2: 2 × ½×F = 2 × 2×D which makes F = 4×D
Hint #7
Substitute (4×D) for F in eq.2b: C = 1½×(4×D) which makes C = 6×D
Hint #8
Substitute (4×D) for F in eq.6b: D + ½×(4×D) = A which becomes D + 2×D = A which makes 3×D = A
Hint #9
Substitute (4×D) for F in eq.6a: D + 2×(4×D) = E which becomes D + 8×D = E which makes 9×D = E
Hint #10
Substitute 6×D for C in eq.4: 6×D + D = B which makes 7×D = B
Solution
Substitute 3×D for A, 7×D for B, 6×D for C, 9×D for E, and 4×D for F in eq.1: 3×D + 7×D + 6×D + D + 9×D + 4×D = 30 which simplifies to 30×D = 30 Divide both sides of the equation above by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 7×D = 7 × 1 = 7 C = 6×D = 6 × 1 = 6 E = 9×D = 9 × 1 = 9 F = 4×D = 4 × 1 = 4 and ABCDEF = 376194