Puzzle for April 23, 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
Help Area
Hint #1
In eq.5, replace C + F with A × D (from eq.3): A × D = (D × E) ÷ A Multiply both sides of the above equation by A: A × A × D = A × (D × E) ÷ A which becomes A × A × D = D × E Since D ≠ 0 (from eq.6), divide both sides by D: (A × A × D) ÷ D = (D × E) ÷ D which becomes eq.5a) A² = E
Hint #2
In eq.1, replace E with A² (from eq.5a): eq.1a) D = A + A²
Hint #3
In eq.6, substitute A² for E (from eq.5a), and (A + A²) for D (from eq.1a): A = (A² × F) ÷ (A + A²) Multiply both sides of the above equation by (A + A²): (A + A²) × A = (A + A²) × (A² × F) ÷ (A + A²) which becomes A² + A³ = A² × F which may be written as A² × (1 + A) = A² × F Since A ≠ 0 (from eq.5), divide both sides by A²: (A² × (1 + A)) ÷ A² = (A² × F) ÷ A² which becomes eq.6a) 1 + A = F
Hint #4
Substitute (A + A²) for D (from eq.1a), and A² for E (from eq.5a) into eq.4: A + A² + A² = A × B which becomes A + 2×A² = A × B Since A ≠ 0 (from eq.5), divide both sides by A: (A + 2×A²) ÷ A = (A × B) ÷ A which becomes eq.4a) 1 + 2×A = B
Hint #5
Substitute A² for E (from eq.5a), 1 + A for F (from eq.6a), and 1 + 2×A for B (from eq.4a) in eq.2: A² - (1 + A) = 1 + 2×A - A² which becomes A² - 1 - A = 1 + 2×A - A² In the equation above, subtract 1 and 2×A from both sides, and add A² to both sides: A² - 1 - A - 1 - 2×A + A² = 1 + 2×A - A² - 1 - 2×A + A² which becomes 2×A² - 2 - 3×A = 0 which may be written as eq.2a) 2×A² - 3×A - 2 = 0
Hint #6
eq.2a is a quadratic equation in standard form. Using the quadratic equation solution formula to solve for A in eq.2a yields: A = { (-1)×(-3) ± sq.rt.[(-3)² - (4 × 2 × (-2))] } ÷ (2 × 2) which becomes A = {3 ± sq.rt.(9 - (-16))} ÷ 4 which becomes A = {3 ± sq.rt.(25)} ÷ 4 which becomes A = (3 ± 5) ÷ 4 In the above equation, either: A = (3 + 5) ÷ 4 = 8 ÷ 4 = 2 or: A = (3 - 5) ÷ 4 = -2 ÷ 4 = -½ Since A must be a non-negative integer, then A ≠ -½ and therefore makes A = 2
Hint #7
Since A = 2, then: B = 1 + 2×A = 1 + 2×2 = 1 + 4 = 5 (from eq.4a) D = A + A² = 2 + 2² = 2 + 4 = 6 (from eq.1a) E = A² = 2² = 4 (from eq.5a) F = 1 + A = 1 + 2 = 3 (from eq.6a)
Solution
Substitute 3 for F, 2 for A, and 6 for D in eq.3: C + 3 = 2 × 6 which becomes C + 3 = 12 Subtract 3 from each side of the equation above: C + 3 - 3 = 12 - 3 which makes C = 9 and makes ABCDEF = 259643