Puzzle for July 5, 2023 ( )
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
eq.6 may be written as: E = (A + C + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (A + C + D) ÷ 3 which becomes eq.6a) 3×E = A + C + D
Hint #2
In eq.6a, replace A + C with D (from eq.2): 3×E = D + D which makes 3×E = 2×D Divide both sides of the above equation by 2: 3×E ÷ 2 = 2×D ÷ 2 which makes eq.6b) 1½×E = D
Hint #3
In eq.3, replace D with 1½×E: 1½×E = B + E Subtract E from each side of the equation above: 1½×E - E = B + E - E which makes ½×E = B Multiply both sides by 2: 2 × ½×E = 2 × B which makes E = 2×B
Hint #4
In eq.6b, substitute (2×B) for E: 1½×(2×B) = D which makes 3×B = D
Hint #5
Substitute B + E for D (from eq.3) into eq.5: E + F = A + B + B + E which becomes E + F = A + 2×B + E Subtract E from each side of the equation above: E + F - E = A + 2×B + E - E which becomes eq.5a) F = A + 2×B
Hint #6
Substitute A + 2×B for F (from eq.5a), and 2×B for E in eq.4: A + 2×B = C + 2×B Subtract 2×B from each side of the above equation: A + 2×B - 2×B = C + 2×B - 2×B which makes A = C
Hint #7
Substitute 3×B for D, and A for C in eq.2: 3×B = A + A which makes 3×B = 2×A Divide both sides of the above equation by 2: 3×B ÷ 2 = 2×A ÷ 2 which makes 1½×B = A and also makes 1½×B = A = C
Hint #8
Substitute 1½×B for A in eq.5a: F = 1½×B + 2×B which makes F = 3½×B
Solution
Substitute 1½×B for A and C, 3×B for D, 2×B for E, and 3½×B for F in eq.1: 1½×B + B + 1½×B + 3×B + 2×B + 3½×B = 25 which simplifies to 12½×B = 25 Divide both sides of the above equation by 12½: 12½×B ÷ 12½ = 25 ÷ 12½ which means B = 2 making A = C = 1½×B = 1½ × 2 = 3 D = 3×B = 3 × 2 = 6 E = 2×B = 2 × 2 = 4 F = 3½×B = 3½ × 2 = 7 and ABCDEF = 323647