Puzzle for August 28, 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
Add C and A to both sides of eq.4: B – C + C + A = D – A + C + A which becomes B + A = D + C which may be written as A + B = C + D In eq.2, replace A + B with C + D: eq.2a) F = C + D
Hint #2
In eq.1, replace A + B with F (from eq.2), and C + D with F (from eq.2a): F + F + E + F = 36 which becomes E + 3×F = 36 Subtract 3×F from both sides of the equation above: E + 3×F – 3×F = 36 – 3×F which becomes eq.1a) E = 36 – 3×F
Hint #3
To make eq.1a true, check several possible values for F and E: If F = 9, then E = 36 – 3×9 = 36 – 27 = 9 If F = 8, then E = 36 – 3×8 = 36 – 24 = 12 If F < 8, then E > 12 Since E must be a one-digit integer, then E = 9 which makes F = 9
Hint #4
In eq.2a, substitute 9 for F: 9 = C + D Subtract D from both sides of the above equation: 9 – D = C + D – D which becomes eq.2b) 9 – D = C
Hint #5
Substitute 9 – D for C (from eq.2b), and 9 for E in eq.3: 9 – D – D + 9 = D which becomes 18 – 2×D = D Add 2×D to each side of the above equation: 18 – 2×D + 2×D = D + 2×D which becomes 18 = 3×D Divide both sides by 3: 18 ÷ 3 = 3×D ÷ 3 which makes 6 = D
Hint #6
Substitute 6 for D in eq.2b: 9 – 6 = C which makes 3 = C
Hint #7
Substitute 6 for D, and 9 for E in eq.5: 6 – A = A + 9 – (6 – A) which is equivalent to 6 – A = A + 9 – 6 + A which becomes 6 – A = 2×A + 3 In the equation above, add A to both sides, and subtract 3 from both sides: 6 – A + A – 3 = 2×A + 3 + A – 3 which makes 3 = 3×A Divide both sides by: 3 ÷ 3 = 3×A ÷ 3 which makes 1 = A
Solution
Substitute 9 for F, and 1 for A in eq.2: 9 = 1 + B Subtract 1 from each side of the equation above: 9 – 1 = 1 + B – 1 which makes 8 = B and makes ABCDEF = 183699