Puzzle for November 16, 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.5, replace C + D with A (from eq.2): A + F = A + B Subtract A from both sides of the above equation: A + F – A = A + B – A which makes F = B
Hint #2
In eq.3, replace B with F: E + F = F Subtract F from both sides of the above equation: E + F – F = F – F which makes E = 0
Hint #3
In eq.4, substitute 0 for E, and add D to both sides: D + 0 + F + D = A + C – D + D which becomes 2×D + F = A + C Add B to both sides of the above equation: 2×D + F + B = A + C + B which may be written as eq.4a) 2×D + F + B = A + B + C
Hint #4
Substitute C + D + F for A + B (from eq.5) in eq.4a: 2×D + F + B = C + D + F + C Subtract D and F from both sides: 2×D + F + B – D – F = C + D + F + C – D – F which simplifies to eq.4b) D + B = 2×C Subtract B from both sides of eq.4b: D + B – B = 2×C – B which becomes eq.4c) D = 2×C – B
Hint #5
Add B to both sides of eq.2: A + B = C + D + B Substitute 2×C for D + B (from eq.4b) in the above equation: A + B = C + 2×C Subtract B from both sides: A + B – B = C + 2×C – B which becomes eq.2a) A = 3×C – B
Hint #6
Substitute 3×C – B for A (from eq.2a), 2×C – B for D (from eq.4c), 0 for E, and B for F in eq.1: 3×C – B + B + C + 2×C – B + 0 + B = 24 which simplifies to 6×C = 24 Divide both sides of the above equation by 6: 6×C ÷ 6 = 24 ÷ 6 which makes C = 4
Hint #7
Substitute 4 for C in eq.2a: A = 3×4 – B which makes eq.2b) A = 12 – B Substitute 4 for C in eq.4c: D = 2×4 – B which makes eq.4d) D = 8 – B
Solution
Substitute 4 for C, and (8 – B) for D (from eq.4d) in eq.6: B + 4 + (8 – B) = 4 × (4 – (8 – B)) which is equivalent to 12 = 4 × (4 – 8 + B) which becomes 12 = 16 – 32 + 4×B which becomes 12 = 4×B – 16 Add 16 to both sides of the equation above: 12 + 16 = 4×B – 16 + 16 which makes 28 = 4×B Divide both sides by 4: 28 ÷ 4 = 4×B ÷ 4 which means 7 = B making A = 12 – B = 12 – 7 = 5 (from eq.2b) D = 8 – B = 8 – 7 = 1 (from eq.4d) F = B = 7 and ABCDEF = 574107