Puzzle for October 22, 2020 ( )
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 F with A + E (from eq.6): B + E = C + A + E Subtract E from both sides of the above equation: B + E – E = C + A + E – E which becomes eq.2a) B = C + A
Hint #2
In eq.3, replace B with C + A for B (from eq.2a): C + E = A + C + A which becomes C + E = 2×A + C Subtract C from both sides of the equation above: C + E – C = 2×A + C – C which makes E = 2×A
Hint #3
In eq.6, substitute 2×A for E: F = A + 2×A which becomes F = 3×A
Hint #4
Substitute 2×A for E, and 3×A for F in eq.5: 2×A + 3×A = B + C + D which becomes eq.5a) 5×A = B + C + D
Hint #5
Substitute 2×A for E in eq.4: D + 2×A = B Subtract 2×A from each side of the equation above: D + 2×A – 2×A = B – 2×A which becomes eq.4a) D = B – 2×A Subtract A from both sides of eq.2a: B – A = C + A – A which becomes eq.2b) B – A = C
Hint #6
Substitute B – A for C (from eq.2b), and B – 2×A for D (from eq.4a) in eq.5a: 5×A = B + B – A + B – 2×A which becomes 5×A = 3×B – 3×A Add 3×A to both sides of the equation above: 5×A + 3×A = 3×B – 3×A + 3×A which makes 8×A = 3×B Divide both sides by 3: 8×A ÷ 3 = 3×B ÷ 3 which makes 2⅔×A = B
Hint #7
Substitute 2⅔×A for B in eq.4a: D = 2⅔×A – 2×A which makes D = ⅔×A
Hint #8
Substitute 2⅔×A for B in eq.2b: 2⅔×A – A = C which makes 1⅔×A = C
Solution
Substitute 2⅔×A for B, 1⅔×A for C, ⅔×A for D, 2×A for E, and 3×A for F in eq.1: A + 2⅔×A + 1⅔×A + ⅔×A + 2×A + 3×A = 33 which simplifies to 11×A = 33 Divide both sides of the above equation by 11: 11×A ÷ 11 = 33 ÷ 11 which means A = 3 making B = 2⅔×A = 2⅔ × 3 = 8 C = 1⅔×A = 1⅔ × 3 = 5 D = ⅔×A = ⅔ × 3 = 2 E = 2×A = 2 × 3 = 6 F = 3×A = 3 × 3 = 9 and ABCDEF = 385269