Puzzle for January 2, 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.4, replace A + B with D + E (from eq.2): E + F = D + E – C Subtract E from both sides of the above equation: E + F – E = D + E – C – E which becomes eq.4a) F = D – C
Hint #2
In eq.3, replace F with D – C (from eq.4a): D – C = C – D Add C and D to both sides of the equation above: D – C + C + D = C – D + C + D which makes 2×D = 2×C Divide both sides by 2: 2×D ÷ 2 = 2×C ÷ 2 which makes D = C
Hint #3
In eq.4a, substitute C for D: F = C – C which makes F = 0
Hint #4
Substitute C for D in eq.6: A – B + C + C = B – C + E which becomes A – B + 2×C = B – C + E Add B and C to each side of the equation above: A – B + 2×C + B + C = B – C + E + B + C which becomes eq.6a) A + 3×C = 2×B + E
Hint #5
Add B to both sides of eq.5: B + B = A – B + E + B which becomes 2×B = A + E Substitute A + E for 2×B in eq.6a: A + 3×C = A + E + E Subtract A from each side of the above equation: A + 3×C – A = A + E + E – A which makes 3×C = 2×E Divide both sides by 2: 3×C ÷ 2 = 2×E ÷ 2 which makes eq.6b) 1½×C = E
Hint #6
Substitute C for D, and 1½×C for E in eq.2: C + 1½×C = A + B which becomes 2½×C = A + B Subtract B from each side of the above equation: 2½×C – B = A + B – B which becomes eq.2a) 2½×C – B = A
Hint #7
Substitute 2½×C – B for A (from eq.2a), and 1½×C for E in eq.6a: 2½×C – B + 3×C = 2×B + 1½×C which becomes 5½×C – B = 2×B + 1½×C In the above equation, add B to both sides, and subtract 1½×C from each side: 5½×C – B + B – 1½×C = 2×B + 1½×C + B – 1½×C which makes 4×C = 3×B Divide both sides by 4: 4×C ÷ 4 = 3×B ÷ 4 which makes C = ¾×B which also makes D = C = ¾×B
Hint #8
Substitute (¾×B) for C in eq.6b: 1½×(¾×B) = E which makes 1⅛×B = E
Hint #9
Substitute (¾×B) for C in eq.2a: 2½×(¾×B) – B = A which becomes 1⅞×B – B = A which makes ⅞×B = A
Solution
Substitute ⅞×B for A, ¾×B for C and D, 1⅛×B for E, and 0 for F in eq.1: ⅞×B + B + ¾×B + ¾×B + 1⅛×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 = ⅞×B = ⅞ × 8 = 7 C = D = ¾×B = ¾ × 8 = 6 E = 1⅛×B = 1⅛ × 8 = 9 and ABCDEF = 786690