Puzzle for February 3, 2022 ( )
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.5, replace E with A + F (from eq.2): D + F = A + B + A + F which becomes D + F = 2×A + B + F Subtract F from each side of the above equation: D + F – F = 2×A + B + F – F which becomes eq.5a) D = 2×A + B
Hint #2
In eq.3, replace D with 2×A + B (from eq.5a): B + F = A + 2×A + B which becomes B + F = 3×A + B Subtract B from both sides of the above equation: B + F – B = 3×A + B – B which makes F = 3×A
Hint #3
In eq.2, substitute 3×A for F: A + 3×A = E which makes 4×A = E
Hint #4
Substitute 3×A for F in eq.3: B + 3×A = A + D Subtract A from each side of the equation above: B + 3×A – A = A + D – A which becomes eq.3a) B + 2×A = D
Hint #5
Substitute 4×A for E, and 3×A for F in eq.6: 4×A + 3×A = B + D which becomes eq.6a) 7×A = B + D
Hint #6
Substitute B + 2×A for D (from eq.3a) in eq.6a: 7×A = B + B + 2×A which becomes 7×A = 2×B + 2×A Subtract 2×A from each side of the above equation: 7×A – 2×A = 2×B + 2×A – 2×A which makes 5×A = 2×B Divide both sides by 2: 5×A ÷ 2 = 2×B ÷ 2 which makes 2½×A = B
Hint #7
Substitute 2½×A for B into eq.3a: 2½×A + 2×A = D which makes 4½×A = D
Hint #8
Substitute 3×A for F, and 2½×A for B in eq.4: C + 3×A = A + 2½×A which becomes C + 3×A = 3½×A Subtract 3×A from both sides of the above equation: C + 3×A – 3×A = 3½×A – 3×A which makes C = ½×A
Solution
Substitute 2½×A for B, ½×A for C, 4½×A for D, 4×A for E, and 3×A for F in eq.1: A + 2½×A + ½×A + 4½×A + 4×A + 3×A = 31 which simplifies to 15½×A = 31 Divide both sides of the above equation by 15½: 15½×A ÷ 15½ = 31 ÷ 15½ which means A = 2 making B = 2½×A = 2½ × 2 = 5 C = ½×A = ½ × 2 = 1 D = 4½×A = 4½ × 2 = 9 E = 4×A = 4 × 2 = 8 F = 3×A = 3 × 2 = 6 and ABCDEF = 251986