Puzzle for April 19, 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 F with A + C (from eq.3): B + C = A + A + C which becomes B + C = 2×A + C Subtract C from each side of the above equation: B + C – C = 2×A + C – C which makes B = 2×A
Hint #2
In eq.2, replace B with 2×A: D = A + 2×A which makes D = 3×A
Hint #3
In eq.6, substitute 2×A for B, 3×A for D, and F for A + C (from eq.3): 2×A + 3×A = F + F which becomes 5×A = 2×F Divide both sides of the above equation by 2: 5×A ÷ 2 = 2×F ÷ 2 which makes 2½×A = F
Hint #4
Substitute 2½×A for F into eq.3: 2½×A = A + C Subtract A from each side of the above equation: 2½×A – A = A + C – A which makes 1½×A = C
Hint #5
Substitute 2×A for B, and 1½×A for C in eq.5: E = A + 2×A + 1½×A which makes E = 4½×A
Solution
Substitute 2×A for B, 1½×A for C, 3×A for D, 4½×A for E, and 2½×A for F in eq.1: A + 2×A + 1½×A + 3×A + 4½×A + 2½×A = 29 which simplifies to 14½×A = 29 Divide both sides of the above equation by 14½: 14½×A ÷ 14½ = 29 ÷ 14½ which means A = 2 making B = 2×A = 2 × 2 = 4 C = 1½×A = 1½ × 2 = 3 D = 3×A = 3 × 2 = 6 E = 4½×A = 4½ × 2 = 9 F = 2½×A = 2½ × 2 = 5 and ABCDEF = 243695