Puzzle for February 24, 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
Add D and C to both sides of eq.4: B - D + D + C = D - C + D + C which becomes B + C = 2×D In the equation above, replace B + C with E (from eq.2): eq.4a) E = 2×D
Hint #2
eq.1 may be written as: C + E + A + B + D + F = 30 In the above equation, replace A + B + D + F with C + E (from eq.6): C + E + C + E = 30 which becomes 2×(C + E) = 30 Divide both sides by 2: 2×(C + E) ÷ 2 = 30 ÷ 2 which becomes eq.1a) C + E = 15
Hint #3
Divide both sides of eq.4a by 2: E ÷ 2 = 2×D ÷ 2 which makes ½×E = D Subtract E from each side of eq.1a: C + E - E = 15 - E which becomes eq.1b) C = 15 - E
Hint #4
In eq.5, substitute ½×E for D, and 15 - E for C (from eq.1b): ½×E + E = 15 - E - ½×E + F which becomes 1½×E = 15 - 1½×E + F In the above equation, add 1½×E to both sides, and subtract 15 from both sides: 1½×E + 1½×E - 15 = 15 - 1½×E + F + 1½×E - 15 which becomes eq.5a) 3×E - 15 = F
Hint #5
Substitute 15 - E for C (from eq.1b) into eq.2: E = B + 15 - E In the above equation, subtract 15 from both sides, and add E to both sides: E - 15 + E = B + 15 - E - 15 + E which becomes eq.2a) 2×E - 15 = B
Hint #6
Substitute 3×E - 15 for F (from eq.5a), 2×E - 15 for B (from eq.2a), and 15 - E for C (from eq.1b) in eq.3: 3×E - 15 = A + 2×E - 15 + 15 - E which becomes 3×E - 15 = A + E Subtract E from each side of the equation above: 3×E - 15 - E = A + E - E which becomes eq.3a) 2×E - 15 = A
Hint #7
Substitute 15 - E for C (from eq.1b), 2×E - 15 for A (from eq.3a) and for B (from eq.2a), ½×E for D, and 3×E - 15 for F (from eq.5a) in eq.6: 15 - E + E = 2×E - 15 + 2×E - 15 + ½×E + 3×E - 15 which becomes 15 = 7½×E - 45 Add 45 to both sides of the above equation: 15 + 45 = 7½×E - 45 + 45 which makes 60 = 7½×E Divide both sides by 7½: 60 ÷ 7½ = 7½×E ÷ 7½ which makes 8 = E
Solution
Since E = 8, then: A = B = 2×E - 15 = 2×8 - 15 = 16 - 15 = 1 (from eq.3a and eq.2a) C = 15 - E = 15 - 8 = 7 (from eq.1b) D = ½×E = ½×8 = 4 F = 3×E - 15 = 3×8 - 15 = 24 - 15 = 9 (from eq.5a) and ABCDEF = 117489