Puzzle for May 23, 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 positive integer.
Scratchpad
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Hint #1
Add B to both sides of eq.5: F + B = C + D – B + B which becomes F + B = C + D In eq.2, replace C + D with F + B: F + B = A + B Subtract B from each side of the above equation: F + B – B = A + B – B which makes F = A
Hint #2
Subtract C, D, and F from both sides of eq.3: B + C – C – D – F = A + D + F – C – D – F which becomes C – D – F = A – B Add the left and right sides of eq.2 to the left and right sides of the equation above, respectively: C – D – F + (C + D) = A – B + (A + B) which becomes eq.3a) 2×C – F = 2×A
Hint #3
In eq.3a, replace F with A: 2×C – A = 2×A Add A to both sides of the equation above: 2×C – A + A = 2×A + A which becomes 2×C = 3×A Divide both sides by 2: 2×C ÷ 2 = 3×A ÷ 2 which makes C = 1½×A
Hint #4
Substitute 1½×A for C, and A for F in eq.4: A + 1½×A – E = E – A Add A and E to both sides of the equation above: A + 1½×A – E + A + E = E – A + A + E which becomes 3½×A = 2×E Divide both sides by 2: 3½×A ÷ 2 = 2×E ÷ 2 which makes 1¾×A = E
Hint #5
Substitute 1¾×A for E, A for F, and 1½×A for C in eq.6: 1¾×A × A = A + (1½×A × D) Divide both sides of the equation above by A: 1¾×A × A ÷ A = (A + (1½×A × D)) ÷ A which becomes eq.6a) 1¾×A = 1 + 1½×D
Hint #6
Subtract 1 from both sides of eq.6a: 1¾×A – 1 = 1 + 1½×D – 1 which becomes 1¾×A – 1 = 1½×D Divide both sides of the above equation by 1½: (1¾×A – 1) ÷ 1½ = 1½×D ÷ 1½ which makes eq.6b) 1⅙×A – ⅔ = D
Hint #7
Substitute 1½×A for C, and 1⅙×A – ⅔ for D (from eq.6b) in eq.2: 1½×A + 1⅙×A – ⅔ = A + B which becomes 2⅔×A – ⅔ = A + B Subtract A from each side of the above equation: 2⅔×A – ⅔ – A = A + B – A which makes eq.2a) 1⅔×A – ⅔ = B
Solution
Substitute 1⅔×A – ⅔ for B (from eq.2a), 1½×A for C, 1⅙×A – ⅔ for D (from eq.6b), 1¾×A for E, and A for F in eq.1: A + 1⅔×A – ⅔ + 1½×A + 1⅙×A – ⅔ + 1¾×A + A = 31 Multiply both sides of the above equation by 12 (to eliminate fractions and simplify calculations): 12 × (A + 1⅔×A – ⅔ + 1½×A + 1⅙×A – ⅔ + 1¾×A + A) = 12 × 31 which becomes 12×A + 20×A – 8 + 18×A + 14×A – 8 + 21×A + 12×A = 372 which simplifies to 97×A – 16 = 372 Add 16 to both sides: 97×A – 16 + 16 = 372 + 16 which makes 97×A = 388 Divide both sides by 97: 97×A ÷ 97 = 388 ÷ 97 which means A = 4 making B = 1⅔×A – ⅔ = 1⅔×4 – ⅔ = 6⅔ – ⅔ = 6 (from eq.2a) C = 1½×A = 1½×4 = 6 D = 1⅙×A – ⅔ = 1⅙×4 – ⅔ = 4⅔ – ⅔ = 4 (from eq.6b) E = 1¾×A = 1¾ × 4 = 7 F = A = 4 and ABCDEF = 466474