Puzzle for May 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
In eq.4, replace B with C + F (from eq.6): C + F + C = A + F which becomes 2×C + F = A + F Subtract F from each side of the equation above: 2×C + F – F = A + F – F which makes 2×C = A
Hint #2
Add E to both sides of eq.5: D – E + E = C + E + E which becomes eq.5a) D = C + 2×E In eq.3, replace D with C + 2×E: E + F = C + C + 2×E which becomes E + F = 2×C + 2×E Subtract E from both sides of the equation above: E + F – E = 2×C + 2×E – E which becomes eq.3a) F = 2×C + E
Hint #3
In eq.6, substitute 2×C + E for F (from eq.3a): C + 2×C + E = B which becomes eq.6a) 3×C + E = B
Hint #4
Substitute 2×C for A, and 3×C + E for B (from eq.6a) in eq.2: 2×C + 3×C + E = D which becomes eq.2a) 5×C + E = D
Hint #5
Substitute 5×C + E for D (from eq.2a) in eq.5a: 5×C + E = C + 2×E Subtract E and C from both sides of the above equation: 5×C + E – E – C = C + 2×E – E – C which makes 4×C = E
Hint #6
Substitute 4×C for E in eq.6a: 3×C + 4×C = B which makes 7×C = B
Hint #7
Substitute 4×C for E in eq.2a: 5×C + 4×C = D which makes 9×C = D
Hint #8
Substitute 4×C for E in eq.3a: F = 2×C + 4×C which makes F = 6×C
Solution
Substitute 2×C for A, 7×C for B, 9×C for D, 4×C for E, and 6×C for F in eq.1: 2×C + 7×C + C + 9×C + 4×C + 6×C = 29 which simplifies to 29×C = 29 Divide both sides of the above equation by 29: 29×C ÷ 29 = 29 ÷ 29 which means C = 1 making A = 2×C = 2 × 1 = 2 B = 7×C = 7 × 1 = 7 D = 9×C = 9 × 1 = 9 E = 4×C = 4 × 1 = 4 F = 6×C = 6 × 1 = 6 and ABCDEF = 271946