Puzzle for March 8, 2023 ( )
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.4, replace B + D with C + F (from eq.3): F - C = C + F - E Subtract F from both sides of the above equation: F - C - F = C + F - E - F which becomes -C = C - E Add C and E to both sides: -C + C + E = C - E + C + E which makes E = 2×C
Hint #2
In eq.5, replace A + C with D + E (from eq.2): B + D + E - A = D + E - D + F which becomes B + D + E - A = E + F In the above equation, subtract E from both sides, and add A to both sides: B + D + E - A - E + A = E + F - E + A which becomes eq.5a) B + D = F + A
Hint #3
In eq.3, replace B + D with F + A (from eq.5a): C + F = F + A Subtract F from each side of the equation above: C + F - F = F + A - F which makes C = A
Hint #4
In eq.2, substitute 2×C for E, and C for A: D + 2×C = C + C which becomes D + 2×C = 2×C Subtract 2×C from each side of the above equation: D + 2×C - 2×C = 2×C - 2×C which makes D = 0
Hint #5
In eq.6, substitute C for A, and 2×C for E: C ÷ C = B - 2×C which becomes 1 = B - 2×C Add 2×C to both sides of the above equation: 1 + 2×C = B - 2×C + 2×C which makes eq.6a) 1 + 2×C = B
Hint #6
Substitute 1 + 2×C for B (from eq.6a), and 0 for D in eq.3: C + F = 1 + 2×C + 0 which becomes C + F = 1 + 2×C Subtract C from both sides of the equation above: C + F - C = 1 + 2×C - C which becomes eq.3a) F = 1 + C
Hint #7
Substitute C for A, 1 + 2×C for B (from eq.6a), 0 for D, 2×C for E, and 1 + C for F (from eq.3a) in eq.1: C + 1 + 2×C + C + 0 + 2×C + 1 + C = 30 which simplifies to 7×C + 2 = 30 Subtract 2 from both sides of the above equation: 7×C + 2 - 2 = 30 - 2 which makes 7×C = 28 Divide both sides by 7: 7×C ÷ 7 = 28 ÷ 7 which means C = 4
Solution
Since C = 4, then: A = C = 4 B = 1 + 2×C = 1 + 2×4 = 1 + 8 = 9 (from eq.6a) E = 2×C = 2×4 = 8 F = 1 + C = 1 + 4 = 5 (from eq.3a) and ABCDEF = 494085