Puzzle for December 11, 2018 ( )
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 B + D (from eq.5): B + B + D - D = A which becomes eq.2a) 2×B = A
Hint #2
Substitute 2×B for A in eq.6: 2×B + B - D = E + F which becomes 3×B - D = E + F From eq.4, substitute B + C + D for E + F in the equation above: 3×B - D = B + C + D Subtract both B and D from each side of the above equation: 3×B - D - B - D = B + C + D - B - D which becomes 2×B - 2×D = C Substitute B + D for C (from eq.5): 2×B - 2×D = B + D Add (2×D - B) to both sides: 2×B - 2×D + (2×D - B) = B + D + (2×D - B) which simplifies to B = 3×D
Hint #3
Replace B with (3×D) in eq.2a: 2×(3×D) = A which makes 6×D = A
Hint #4
Replace B with 3×D in eq.5: 3×D + D = C which makes 4×D = C
Hint #5
Substitute 6×D for A, and 4×D for C in eq.3: 4×D - 6×D = 6×D - F which becomes -2×D = 6×D - F Add F + 2×D to both sides of the above equation: -2×D + F + 2×D = 6×D - F + F + 2×D which simplifies to F = 8×D
Hint #6
Substitute 6×D for A, 3×D for B, and 8×D for F in eq.6: 6×D + 3×D - D = E + 8×D which becomes 8×D = E + 8×D Subtract 8×D from both sides of the above equation: 8×D - 8×D = E + 8×D - 8×D which means 0 = E
Solution
Substitute 6×D for A, 3×D for B, 4×D for C, 0 for E, and 8×D for F in eq.1: 6×D + 3×D + 4×D + D + 0 + 8×D = 22 which simplifies to 22×D = 22 Divide both sides by 22: 22×D ÷ 22 = 22 ÷ 22 which means D = 1 making A = 6×D = 6 × 1 = 6 B = 3×D = 3 × 1 = 3 C = 4×D = 4 × 1 = 4 F = 8×D = 8 × 1 = 8 and ABCDEF = 634108