Puzzle for September 27, 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 positive integer.
Scratchpad
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Hint #1
Add B and D to both sides of eq.3: F – D + B + D = D – B + B + D which becomes eq.3a) F + B = 2×D
Hint #2
Add A to both sides of eq.5: D – A + A = A – B + A which becomes eq.5a) D = 2×A – B
Hint #3
In eq.2, replace C with A + E (from eq.4): E + F = B + A + E Subtract E from both sides of the above equation: E + F – E = B + A + E – E which becomes eq.2a) F = B + A
Hint #4
Substitute B + A for F (from eq.2a), and (2×A – B) for D (from eq.5a) in eq.3a: B + A + B = 2×(2×A – B) which is equivalent to A + 2×B = 4×A – 2×B In the above equation, add 2×B to both sides, and subtract A from each side: A + 2×B + 2×B – A = 4×A – 2×B + 2×B – A which simplifies to 4×B = 3×A Divide both sides by 4: 4×B ÷ 4 = 3×A ÷ 4 which makes B = ¾×A
Hint #5
Substitute ¾×A for B in eq.2a: F = ¾×A + A which makes F = 1¾×A
Hint #6
Substitute ¾×A for B in eq.5a: D = 2×A – ¾×A which makes D = 1¼×A
Hint #7
Substitute ¾×A for B, and 1¼×A for D in eq.6: ¾×A × E = A + 1¼×A which becomes ¾×A × E = 2¼×A Divide both sides by ¾×A: ¾×A × E ÷ ¾×A = 2¼×A ÷ ¾×A which makes E = 3
Hint #8
Substitute 3 for E in eq.4: eq.4a) C = A + 3
Solution
Substitute ¾×A for B, A + 3 for C (from eq.4a), 1¼×A for D, 3 for E, and 1¾×A for F in eq.1: A + ¾×A + A + 3 + 1¼×A + 3 + 1¾×A = 29 which simplifies to 5¾×A + 6 = 29 Subtract 6 from each side: 5¾×A + 6 – 6 = 29 – 6 which makes 5¾×A = 23 Divide both sides by 5¾: 5¾×A ÷ 5¾ = 23 ÷ 5¾ which means A = 4 making B = ¾×A = ¾ × 4 = 3 C = A + 3 = 4 + 3 = 7 (from eq.4a) D = 1¼×A = 1¼ × 4 = 5 F = 1¾×A = 1¾ × 4 = 7 and ABCDEF = 437537