Puzzle for June 2, 2020 ( )
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.6, replace E + F with B + C (from eq.2): A + D = B + B + C which becomes eq.6a) A + D = 2×B + C
Hint #2
In eq.6a, replace C with A + B (from eq.5): A + D = 2×B + A + B which becomes A + D = 3×B + A Subtract A from each side of the above equation: A + D – A = 3×B + A – A which makes D = 3×B
Hint #3
In eq.4, substitute 3×B for D: F = B + 3×B which makes F = 4×B
Hint #4
Substitute 3×B for D, 4×B for F, and A + B for C (from eq.5) in eq.3: B + 3×B + 4×B = A + A + B which becomes 8×B = 2×A + B Subtract B from both sides of the equation above: 8×B – B = 2×A + B – B which makes 7×B = 2×A Divide both sides by 2: 7×B ÷ 2 = 2×A ÷ 2 which makes 3½×B = A
Hint #5
Substitute 3½×B for A in eq.5: C = 3½×B + B which makes C = 4½×B
Hint #6
Substitute 4×B for F, and 4½×B for C in eq.2: E + 4×B = B + 4½×B which becomes E + 4×B = 5½×B Subtract 4×B from each side of the above equation: E + 4×B – 4×B = 5½×B – 4×B which makes E = 1½×B
Solution
Substitute 3½×B for A, 4½×B for C, 3×B for D, 1½×B for E, and 4×B for F in eq.1: 3½×B + B + 4½×B + 3×B + 1½×B + 4×B = 35 which simplifies to 17½×B = 35 Divide both sides of the above equation by 17½: 17½×B ÷ 17½ = 35 ÷ 17½ which means B = 2 making A = 3½×B = 3½ × 2 = 7 C = 4½×B = 4½ × 2 = 9 D = 3×B = 3 × 2 = 6 E = 1½×B = 1½ × 2 = 3 F = 4×B = 4 × 2 = 8 and ABCDEF = 729638