Puzzle for November 7, 2022 ( )
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.
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Hint #1
In eq.2, replace A with C + E (from eq.2): C + E = D + E Subtract E from each side of the equation above: C + E - E = D + E - E which makes C = D
Hint #2
In eq.4, replace D + E with A (from eq.2): F = A + A which makes eq.4a) F = 2×A
Hint #3
In eq.5, substitute 2×A for F: 2×A - C = A + C In the above equation, add C to both sides, and subtract A from both sides: 2×A - C + C - A = A + C + C - A which makes A = 2×C
Hint #4
Substitute (2×C) for A in eq.4a: F = 2×(2×C) which makes F = 4×C
Hint #5
Substitute 2×C for A in eq.3: 2×C = C + E Subtract C from each side of the above equation: 2×C - C = C + E - C which makes C = E
Hint #6
Substitute 2×C for A, and C for D in eq.6: 2×C - B = B + C In the above equation, add B to both sides, and subtract C from both sides: 2×C - B + B - C = B + C + B - C which makes C = 2×B Divide both sides of the above equation by 2: C ÷ 2 = 2×B ÷ 2 which makes ½×C = B
Solution
Substitute 2×C for A, ½×C for B, C for D and E, and 4×C for F in eq.1: 2×C + ½×C + C + C + C + 4×C = 19 which simplifies to 9½×C = 19 Divide both sides of the above equation by 9½: 9½×C ÷ 9½ = 19 ÷ 9½ which means C = 2 making A = 2×C = 2 × 2 = 4 B = ½×C = ½ × 2 = 1 D = E = C = 2 F = 4×C = 4 × 2 = 8 and ABCDEF = 412228