Puzzle for April 14, 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.6, replace D + E with C + F (from eq.4): C + F + F = A + C which becomes C + 2×F = A + C Subtract C from both sides of the above equation: C + 2×F - C = A + C - C which makes eq.6a) 2×F = A
Hint #2
In eq.4, replace E with C + D (from eq.2): C + F = D + C + D which becomes C + F = C + 2×D Subtract C from both sides of the above equation: C + F - C = C + 2×D - C which makes F = 2×D
Hint #3
In eq.6a, substitute (2×D) for F: 2×(2×D) = A which makes 4×D = A
Hint #4
Substitute 4×D for A, and 2×D for F in eq.5: 4×D + 2×D = B + E which becomes 6×D = B + E Subtract B from both sides of the above equation: 6×D - B = B + E - B which becomes eq.5a) 6×D - B = E
Hint #5
Substitute B + F for C (from eq.3) in eq.2: eq.2a) E = B + F + D
Hint #6
Substitute 6×D - B for E (from eq.5a), and 2×D for F in eq.2a: 6×D - B = B + 2×D + D which becomes 6×D - B = B + 3×D In the above equation, add B to both sides, and subtract 3×D from both sides: 6×D - B + B - 3×D = B + 3×D + B - 3×D which makes 3×D = 2×B Divide both sides by 2: 3×D ÷ 2 = 2×B ÷ 2 which makes 1½×D = B
Hint #7
Substitute 1½×D for B in eq.5a: 6×D - 1½×D = E which makes 4½×D = E
Hint #8
Substitute 1½×D for B, and 2×D for F in eq.3: 1½×D + 2×D = C which makes 3½×D = C
Solution
Substitute 4×D for A, 1½×D for B, 3½×D for C, 4½×D for E, and 2×D for F in eq.1: 4×D + 1½×D + 3½×D + D + 4½×D + 2×D = 33 which simplifies to 16½×D = 33 Divide both sides of the equation above by 16½: 16½×D ÷ 16½ = 33 ÷ 16½ which means D = 2 making A = 4×D = 4 × 2 = 8 B = 1½×D = 1½ × 2 = 3 C = 3½×D = 3½ × 2 = 7 E = 4½×D = 4½ × 2 = 9 F = 2×D = 2 × 2 = 4 and ABCDEF = 837294