Puzzle for September 4, 2019 ( )
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.3, replace B + C with F (from eq.2): F + D = A + E In eq.5, replace F + D with A + E: A + B = A + E Subtract A from both sides of the above equation: A + B – A = A + E – A which makes B = E
Hint #2
In eq.4, substitute B for E: D = B + B which makes D = 2×B
Hint #3
Substitute 2×B for D, and B for E in eq.3: B + C + 2×B = A + B which becomes C + 3×B = A + B Subtract B from both sides of the above equation: C + 3×B – B = A + B – B which becomes eq.3a) C + 2×B = A
Hint #4
Substitute B for E, B + C for F (from eq.2), and C + 2×B for A (from eq.3a) in eq.6: C – B + B + C = C + 2×B + B – C which simplifies to 2×C = 3×B Divide both sides of the equation above by 2: 2×C ÷ 2 = 3×B ÷ 2 which makes C = 1½×B
Hint #5
Substitute 1½×B for C in eq.3a: 1½×B + 2×B = A which means 3½×B = A
Hint #6
Substitute 1½×B for C in eq.2: F = B + 1½×B which makes F = 2½×B
Solution
Substitute 3½×B for A, 1½×B for C, 2×B for D, B for E, and 2½×B for F in eq.1: 3½×B + B + 1½×B + 2×B + B + 2½×B = 23 which simplifies to 11½×B = 23 Divide both sides of the equation above by 11½: 11½×B ÷ 11½ = 23 ÷ 11½ which means B = 2 making A = 3½×B = 3½ × 2 = 7 C = 1½×B = 1½ × 2 = 3 D = 2×B = 2 × 2 = 4 E = B = 2 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 723425