Puzzle for October 7, 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
Add E to both sides of eq.4: D + E + F + E = A + B – C – E + E which becomes eq.4a) D + 2×E + F = A + B – C Add D, E, and A to both sides of eq.5: B + C – D – E + D + E + A = E + F – A + D + E + A which becomes B + C + A = 2×E + F + D which may be written as eq.5a) A + B + C = D + 2×E + F
Hint #2
In eq.4a, replace D + 2×E + F with A + B + C (from eq.5a): A + B + C = A + B – C Subtract A, B, and C from each side of the above equation: A + B + C – A – B – C = A + B – C – A – B – C which simplifies to 0 = –2×C which means 0 = C
Hint #3
In eq.2, replace C with 0: F = A + 0 which makes F = A
Hint #4
In eq.3, substitute 0 for C: 0 – D + F = D + E which becomes –D + F = D + E Add D to both sides of the equation above: –D + F + D = D + E + D which becomes eq.3a) F = 2×D + E
Hint #5
eq.6 may be written as: E = (C + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (C + D + F) ÷ 3 which becomes eq.6a) 3×E = C + D + F
Hint #6
Substitute 0 for C, and 2×D + E for F (from eq.3a) in eq.6a: 3×E = 0 + D + 2×D + E which becomes 3×E = 3×D + E Subtract E from each side of the equation above: 3×E – E = 3×D + E – E which makes 2×E = 3×D Divide both sides by 2: 2×E ÷ 2 = 3×D ÷ 2 which makes E = 1½×D
Hint #7
Substitute 1½×D for E in eq.3a: F = 2×D + 1½×D which makes F = 3½×D and also makes A = F = 3½×D
Hint #8
Substitute 0 for C, 1½×D for E, and A for F in eq.5: B + 0 – D – 1½×D = 1½×D + A – A which becomes B – 2½×D = 1½×D Add 2½×D to both sides of the equation above: B – 2½×D + 2½×D = 1½×D + 2½×D which makes B = 4×D
Solution
Substitute 3½×D for A and F, 4×D for B, 0 for C, and 1½×D for E in eq.1: 3½×D + 4×D + 0 + D + 1½×D + 3½×D = 27 which simplifies to 13½×D = 27 Divide both sides of the above equation by 13½: 13½×D ÷ 13½ = 27 ÷ 13½ which means D = 2 making A = F = 3½×D = 3½ × 2 = 7 B = 4×D = 4 × 2 = 8 E = 1½×D = 1½ × 2 = 3 and ABCDEF = 780237