Puzzle for March 20, 2019 ( )
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.2, replace C with D + E (from eq.6): A + E = D + E + F Subtract E from each side of the above equation: A + E - E = D + E + F - E which becomes A = D + F In eq.3, replace D + F with A: eq.3a) B + C = A
Hint #2
In eq.5, replace C with D + E (from eq.6): A + B - F = D + E - D which becomes A + B - F = E Add F to both sides of the equation above: A + B - F + F = E + F which becomes eq.5a) A + B = E + F
Hint #3
In eq.4, replace A + B with E + F (fom eq.5a): C + E = E + F Subtract E from both sides of the equation above: C + E - E = E + F - E which makes C = F
Hint #4
In eq.3, substitute C for F: B + C = D + C Subtract C from each side of the above equation: B + C - C = D + C - C which makes B = D
Hint #5
Substitute C for F, and B for D in eq.5: A + B - C = C - B Add (C - B) to both sides of the above equation: A + B - C + (C - B) = C - B + (C - B) which becomes A = 2×C - 2×B Substitute 2×C - 2×B for A in eq.3a: B + C = 2×C - 2×B Add (2×B - C) to both sides: B + C + (2×B - C) = 2×C - 2×B + (2×B - C) which simplifies to 3×B = C and which also makes F = C = 3×B
Hint #6
Substitute 3×B for C in eq.3a: B + 3×B = A which makes 4×B = A
Hint #7
Substitute B for D, and 3×B for C in eq.6: B + E = 3×B Subtract B from each side: B + E - B = 3×B - B which makes E = 2×B
Solution
Substitute 4×B for A, 3×B for C and F, B for D, and 2×B for E in eq.1: 4×B + B + 3×B + B + 2×B + 3×B = 14 which simplifies to 14×B = 14 Divide both sides by 14: 14×B ÷ 14 = 14 ÷ 14 which means B = 1 making A = 4×B = 4 × 1 = 4 C = F = 3×B = 3 × 1 = 3 D = B = 1 E = 2×B = 2 × 1 = 2 and ABCDEF = 413123