Puzzle for July 14, 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
eq.4 may be re-written as: A + F + D = C In the above equation, replace A + F with E – D (from eq.5): E – D + D = C which becomes E = C
Hint #2
In eq.2, replace E with C: C + D = A + C Subtract C from both sides of the above equation: C + D – C = A + C – C which makes A = D
Hint #3
In eq.6, substitute C for E, and B + D for F (from eq.3): C – B + C = B + B + D which becomes 2×C – B = 2×B + D Add B to both sides of the equation above: 2×C – B + B = 2×B + D + B which becomes 2×C = 3×B + D Divide both sides by 2: 2×C ÷ 2 = (3×B + D) ÷ 2 which becomes eq.6a) C = 1½×B + ½×D
Hint #4
Substitute D for A, and B + D for F (from eq.3), and 1½×B + ½×D for C (from eq.6a) in eq.4: D + D + B + D = 1½×B + ½×D which becomes B + 3×D = 1½×B + ½×D Subtract B and ½×D from both sides of the equation above: B + 3×D – B – ½×D = 1½×B + ½×D – B – ½×D which makes 2½×D = ½×B Multiply both sides by 2: 2½×D × 2 = ½×B × 2 which makes 5×D = B
Hint #5
Substitute 5×D for B in eq.6a: F = 5×D + D which makes F = 6×D
Hint #6
Substitute (5×D) for B in eq.6a: C = 1½×(5×D) + ½×D which is equivalent to C = 7½×D + ½×D which makes C = 8×D and also makes E = C = 8×D
Solution
Substitute D for A, 5×D for B, 8×D for C and E, and 6×D for F in eq.1: D + 5×D + 8×D + D + 8×D + 6×D = 29 which simplifies to 29×D = 29 Divide both sides of the equation above by 29: 29×D ÷ 29 = 29 ÷ 29 which means D = 1 making A = D = 1 B = 5×D = 5 × 1 = 5 C = E = 8×D = 8 × 1 = 8 F = 6×D = 6 × 1 = 6 and ABCDEF = 158186