Puzzle for August 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
Add F to both sides of eq.5: A + E + F = B + D + F In the equation above, replace E + F with C + D (from eq.4): A + C + D = B + D + F Subtract C and D from both sides: A + C + D – C – D = B + D + F – C – D which becomes eq.5a) A = B + F – C
Hint #2
In eq.2, replace A with B + F – C (from eq.5a): B + C = B + F – C + F In the above equation, subtract B from both sides, and add C to both sides: B + C – B + C = B + F – C + F – B + C which simplifies to 2×C = 2×F Divide both sides by 2: 2×C ÷ 2 = 2×F ÷ 2 which means C = F
Hint #3
In eq.5a, substitute C for F: A = B + C – C which means A = B
Hint #4
Substitute A for B in eq.5: A + E = A + D Subtract A from both sides of the above equation: A + E – A = A + D – A which means E = D
Hint #5
Substitute C for F, and A for B in eq.3: C + C – A = A + A which becomes 2×C – A = 2×A Add A to both sides of the equation above: 2×C – A + A = 2×A + A which means 2×C = 3×A Divide both sides by 2: 2×C ÷ 2 = 3×A ÷ 2 which makes C = 1½×A and also makes eq.3a) F = C = 1½×A
Hint #6
Substitute D for E, 1½×A for C and F, and A for B in eq.6: D + D = 1½×A + 1½×A – (D – A) which becomes 2×D = 3×A – D + A Add D to both sides of the equation above: 2×D + D = 3×A – D + A + D which becomes 3×D = 4×A Divide both sides by 4: 3×D ÷ 4 = 4×A ÷ 4 which makes ¾×D = A and also makes B = A = ¾×D
Hint #7
Substitute ¾×D for A in eq.3a: F = C = 1½×(¾×D) which means F = C = 1⅛×D
Solution
Substitute ¾×D for A and B, 1⅛×D for C and F, and D for E in eq.1: ¾×D + ¾×D + 1⅛×D + D + D + 1⅛×D = 46 which simplifies to 5¾×D = 46 Divide both sides of the equation above by 5¾: 5¾×D ÷ 5¾ = 46 ÷ 5¾ which means D = 8 making A = B = ¾×D = ¾ × 8 = 6 C = F = 1⅛×D = 1⅛ × 8 = 9 D = E = 8 and ABCDEF = 669889