Puzzle for February 11, 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
In eq.3, replace E + F with A + D (from eq.5): C + D = B + A + D Subtract D from each side of the equation above: C + D – D = B + A + D – D which becomes C = B + A which is the same as eq.3a) C = A + B
Hint #2
In eq.6, replace A + B with C (from eq.3a): F = C + C which makes eq.6a) F = 2×C
Hint #3
In eq.3, substitute A for B + E (from eq.2), and 2×C for F: C + D = A + 2×C Subtract C from both sides of the equation above: C + D – C = A + 2×C – C which becomes eq.3b) D = A + C
Hint #4
Substitute A + C for D (from eq.3b) in eq.4: A + C – E = C + E In the above equation, subtract C from each side, and add E to each side: A + C – E – C + E = C + E – C + E which makes A = 2×E
Hint #5
Substitute 2×E for A in eq.2: B + E = 2×E Subtract E from both sides of the above equation: B + E – E = 2×E – E which makes B = E
Hint #6
Substitute 2×C for F, and A + C for D (from eq.3b) in eq.5: E + 2×C = A + A + C which becomes E + 2×C = 2×A + C Subtract C from each side of the equation above: E + 2×C – C = 2×A + C – C which becomes eq.5a) E + C = 2×A
Hint #7
Substitute (2×E) for A in eq.5a: E + C = 2×(2×E) which becomes E + C = 4×E Subtract C from both sides of the equation above: E + C – E = 4×E – E which makes C = 3×E
Hint #8
Substitute (3×E) for C in eq.6a: F = 2×(3×E) which makes F = 6×E
Hint #9
Substitute 2×E for A, and 3×E for C in eq.3b: D = 2×E + 3×E which makes D = 5×E
Solution
Substitute 2×E for A, E for B, 3×E for C, 5×E for D, and 6×E for F in eq.1: 2×E + E + 3×E + 5×E + E + 6×E = 18 which simplifies to 18×E = 18 Divide both sides of the equation above by 18: 18×E ÷ 18 = 18 ÷ 18 which means E = 1 making A = 2×E = 2 × 1 = 2 B = E = 1 C = 3×E = 3 × 1 = 3 D = 5×E = 5 × 1 = 5 F = 6×E = 6 × 1 = 6 and ABCDEF = 213516