Puzzle for September 5, 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
eq.1 may be re-written as: A + E + F + B + D + C = 35 In the above equation, replace A + E + F with C + D (from eq.3), and replace B + D with C (from eq.2): C + D + C + C = 35 which becomes 3×C + D = 35 Subtract 3×C from each side: 3×C + D – 3×C = 35 – 3×C which becomes eq.1a) D = 35 – 3×C
Hint #2
To make eq.1a true, check several possible values for C and D: If C = 9, then D = 35 – 3×9 = 35 – 27 = 8 If C = 8, then D = 35 – 3×8 = 35 – 24 = 11 If C < 8, then D > 11 Since D must be a one-digit integer, then D = 8 which makes C = 9
Hint #3
Substitute 8 for D, and 9 for C in eq.2: B + 8 = 9 Subtract 8 from both sides of the equation above: B + 8 – 8 = 9 – 8 which makes B = 1
Hint #4
Substitute 1 for B, and 8 for D in eq.5: A – 1 + F = 1 + 8 Add 1 to both sides of the above equation: A – 1 + F + 1 = 1 + 8 + 1 which becomes eq.5a) A + F = 10
Hint #5
eq.3 may be written as: C + D = A + F + E Substitute 9 for C, 8 for D, and 10 for A + F (from eq.5a) in eq.3: 9 + 8 = 10 + E Subtract 10 from both sides: 9 + 8 – 10 = 10 + E – 10 which means 7 = E
Hint #6
Substitute 7 for E, and 1 for B in eq.4: 7 – 1 = A + 1 Subtract 1 from both sides: 7 – 1 – 1 = A + 1 – 1 which makes 5 = A
Solution
Substitute 5 for A in eq.5a: 5 + F = 10 Subtract 5 from both sides: 5 + F – 5 = 10 – 5 which means F = 5 and ABCDEF = 519875