Puzzle for September 7, 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
Add D and E to both sides of eq.2: C – E + D + E = B – D + D + E which becomes C + D = B + E In eq.4, replace B + E with C + D: A + D = C + D Subtract D from both sides of the above equation: A + D – D = C + D – D which makes A = C
Hint #2
Add A and E to each side of eq.3: D – E + F + A + E = B – A + A + E which becomes D + F + A = B + E In eq.4, replace B + E with D + F + A: A + D = D + F + A Subtract both A and D from each side of the equation above: A + D – A – D = D + F + A – A – D which simplifies to 0 = F
Hint #3
Add both C and E to each side of eq.6: E – C + C + E = B – E + C + E which becomes eq.6a) 2×E = B + C Add both D and E to each side of eq.5: B + D – E + D + E = A + C – D + E + D + E which becomes eq.5a) B + 2×D = A + C + 2×E
Hint #4
In eq.5a, substitute B + C for 2×E (from eq.6a): B + 2×D = A + C + B + C which becomes B + 2×D = A + B + 2×C In the above equation, substitute C for A, and subtract B from both sides: B + 2×D – B = C + B + 2×C – B which becomes 2×D = 3×C Divide both sides by 2: 2×D ÷ 2 = 3×C ÷ 2 which makes eq.5b) D = 1½×C
Hint #5
Substitute 1½×C for D in eq.2: C – E = B – 1½×C Subtract C from both sides of the above equation: C – E – C = B – 1½×C – C which becomes –E = B – 2½×C Multiply both sides by (–1): –E × (–1) = (B – 2½×C) × (–1) which becomes E = –B + 2½×C which is equivalent to eq.2a) E = 2½×C – B
Hint #6
Substitute (2½×C – B) for E (from eq.2a) in eq.6a: 2×(2½×C – B) = B + C which becomes 5×C – 2×B = B + C In the above equation, add 2×B to each side, and subtract C from both sidess: 5×C – 2×B + 2×B – C = B + C + 2×B – C which makes 4×C = 3×B Divide both sides by 4: 4×C ÷ 4 = 3×B ÷ 4 which means C = ¾×B and also means A = C = ¾×B
Hint #7
Substitute (¾×B) for C in eq.5b: D = 1½×(¾×B) which makes D = 1⅛×B
Hint #8
Substitute (¾×B) for C in eq.2a: E = 2½×(¾×B) – B which becomes E = 1⅞×B – B which makes E = ⅞×B
Solution
Substitute ¾×B for A and C, 1⅛×B for D, ⅞×B for E, and 0 for F in eq.1: ¾×B + B + ¾×B + 1⅛×B + ⅞×B + 0 = 36 which simplifies to 4½×B = 36 Divide both sides of the equation above by 4½: 4½×B ÷ 4½ = 36 ÷ 4½ which means B = 8 making A = C = ¾×B = ¾ × 8 = 6 D = 1⅛×B = 1⅛ × 8 = 9 E = ⅞×B = ⅞ × 8 = 7 and ABCDEF = 686970