Puzzle for August 31, 2022 ( )
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
In eq.6, replace E + F with A (from eq.2): D + A = A + B – E In the above equation, subtract A from each side, and add E to both sides: D + A – A + E = A + B – E – A + E which becomes eq.6a) D + E = B
Hint #2
In eq.3, replace B with D + E (from eq.6a): C + E = D + E + D which becomes C + E = 2×D + E Subtract E from each side of the equation above: C + E – E = 2×D + E – E which makes C = 2×D
Hint #3
eq.5 may be written as: B + E = A + F + D In the above equation, substitute B + C for A + F (from eq.4): B + E = B + C + D Subtract B from both sides: B + E – B = B + C + D – B which becomes eq.5a) E = C + D
Hint #4
Substitute 2×D for C in eq.5a: E = 2×D + D which makes E = 3×D
Hint #5
Substitute 3×D for E in eq.6a: D + 3×D = B which makes 4×D = B
Hint #6
In eq.5, replace A with E + F (from eq.2), and D with B – E (from eq.6a): B + E = E + F + B – E + F which becomes B + E = 2×F + B Subtract B from both sides of the equation above: B + E – B = 2×F + B – B which becomes eq.5b) E = 2×F
Hint #7
Substitute 3×D for E in eq.5b: 3×D = 2×F Divide both sides of the above equation by 2: 3×D ÷ 2 = 2×F ÷ 2 which makes 1½×D = F
Hint #8
In eq.2, substitute 3×D for E, and 1½×D for F: 3×D + 1½×D = A which makes 4½×D = A
Solution
Substitute 4½×D for A, 4×D for B, 2×D for C, 3×D for E, and 1½×D for F in eq.1: 4½×D + 4×D + 2×D + D + 3×D + 1½×D = 32 which simplifies to 16×D = 32 Divide both sides of the above equation by 16: 16×D ÷ 16 = 32 ÷ 16 which means D = 2 making A = 4½×D = 4½ × 2 = 9 B = 4×D = 4 × 2 = 8 C = 2×D = 2 × 2 = 4 E = 3×D = 3 × 2 = 6 F = 1½×D = 1½ × 2 = 3 and ABCDEF = 984263