Puzzle for September 23, 2021 ( )
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 B, E, and D to both sides of eq.5: F – B – E + B + E + D = A – D + B + E + D which becomes F + D = A + B + E which may be written as eq.5a) D + F = A + E + B Add E to both sides of eq.4: D – E + F + E = B + E + E which becomes eq.4a) D + F = B + 2×E
Hint #2
In eq.5a, replace D + F with B + 2×E (from eq.4a): B + 2×E = A + E + B Subtract B and E from each side of the equation above: B + 2×E – B – E = A + E + B – B – E which becomes E = A
Hint #3
In eq.5a, replace A + E with B + F (from eq.2): D + F = B + F + B which becomes D + F = 2×B + F Subtract F from both sides of the above equation: D + F – F = 2×B + F – F which makes D = 2×B
Hint #4
In eq.6, substitute A for E: A + F – C = A + B + C In the equation above, subtract A from both sides, and add C to both sides: A + F – C – A + C = A + B + C – A + C which becomes eq.6a) F = B + 2×C
Hint #5
Substitute A for E, and B + 2×C for F (from eq.6a) in eq.2: A + A = B + B + 2×C which becomes 2×A = 2×B + 2×C Divide both sides of the above equation by 2: 2×A ÷ 2 = (2×B + 2×C) ÷ 2 which becomes eq.2a) A = B + C
Hint #6
Substitute B + C for A (from eq.2a), and 2×B for D in eq.3: C + F = B + C + B + 2×B which becomes C + F = 4×B + C Subtract C from each side of the equation above: C + F – C = 4×B + C – C which makes F = 4×B
Hint #7
Substitute 4×B for F in eq.6a: 4×B = B + 2×C Subtract B from both sides of the equation above: 4×B – B = B + 2×C – B which becomes 3×B = 2×C Divide both sides by 2: 3×B ÷ 2 = 2×C ÷ 2 which makes 1½×B = C
Hint #8
Substitute 1½×B for C in eq.2a: A = B + 1½×B which makes A = 2½×B and also makes E = A = 2½×B
Solution
Substitute 2½×B for A and E, 1½×B for C, 2×B for D, and 4×B for F in eq.1: 2½×B + B + 1½×B + 2×B + 2½×B + 4×B = 27 which simplifies to 13½×B = 27 Divide both sides of the above equation by 13½: 13½×B ÷ 13½ = 27 ÷ 13½ which means B = 2 making A = E = 2½×B = 2½ × 2 = 5 C = 1½×B = 1½ × 2 = 3 D = 2×B = 2 × 2 = 4 F = 4×B = 4 × 2 = 8 and ABCDEF = 523458