Puzzle for October 8, 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.4, replace D with B + E (from eq.2): C + B + E = A + B + F Subtract B from each side of the above equation: C + B + E – B = A + B + F – B which becomes eq.4a) C + E = A + F
Hint #2
In eq.5, replace D with B + E (from eq.2), and C + E with A + F (from eq.4a): B + E – E + F = A + F which becomes B + F = A + F Subtract F from both sides of the above equation: B + F – F = A + F – F which makes B = A
Hint #3
In eq.4, substitute A + B + C for F (from eq.3): C + D = A + B + A + B + C which becomes C + D = 2×A + 2×B + C Subtract C from each side of the equation above: C + D – C = 2×A + 2×B + C – C which becomes eq.4b) D = 2×A + 2×B
Hint #4
Substitute A for B in eq.4b: D = 2×A + 2×A which makes D = 4×A
Hint #5
Substitute A for B, and 4×A for D in eq.2: A + E = 4×A Subtract A from both sides of the equation above: A + E – A = 4×A – A which makes E = 3×A
Hint #6
Substitute A for B in eq.3: F = A + A + C which becomes eq.3a) F = 2×A + C
Hint #7
eq.6 may be written as: E = (B + C + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × (B + C + D + F) ÷ 4 which becomes eq.6a) 4×E = B + C + D + F
Hint #8
In eq.6a, substitute (3×A) for E, A for B, 4×A for D, and 2×A + C for F (from eq.3a): 4×(3×A) = A + C + 4×A + 2×A + C which becomes 12×A = 7×A + 2×C Subtract 7×A from both sides of the equation above: 12×A – 7×A = 7×A + 2×C – 7×A which makes 5×A = 2×C Divide both sides by 2: 5×A ÷ 2 = 2×C ÷ 2 which makes 2½×A = C
Hint #9
Substitute 2½×A for C in eq.3a: F = 2×A + 2½×A which makes F = 4½×A
Solution
Substitute A for B, 2½×A for C, 4×A for D, 3×A for E, and 4½×A in F in eq.1: A + A + 2½×A + 4×A + 3×A + 4½×A = 32 which simplifies to 16×A = 32 Divide both sides of the above equation by 16: 16×A ÷ 16 = 32 ÷ 16 which means A = 2 making B = A = 2 C = 2½×A = 2½ × 2 = 5 D = 4×A = 4 × 2 = 8 E = 3×A = 3 × 2 = 6 F = 4½×A = 4½ × 2 = 9 and ABCDEF = 225869