Puzzle for November 8, 2022 ( )
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.
Once again, our thanks go out to frequent contributor Judah S (age 16), for sending us this puzzle. Thank you, Judah!
Scratchpad
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Hint #1
In eq.4, substitute (B + D) for E (from eq.2): D - (B + D) = B - D which becomes D - B - D = B - D which becomes -B = B - D And B and D to both sides of the above equation: -B + B + D = B - D + B + D which makes D = 2×B
Hint #2
In eq.2, replace D with 2×B: E = B + 2×B which makes E = 3×B
Hint #3
In eq.6, replace D with 2×B, and E with 3×B: 2×B + 3×B = B × C which becomes 5×B = B × C Divide both sides of the above equation by B: 5×B ÷ B = (B × C) ÷ B which makes 5 = C
Hint #4
Substitute 5 for C, and 2×B for D in eq.3: eq.3a) F = 5 + 2×B
Hint #5
In eq.1, substitute 5 for C: 5 = A + B Subtract B from each side of the equation above: 5 - B = A + B - B which becomes eq.1a) 5 - B = A
Hint #6
Substitute 3×B for E, 5 + 2×B for F (from eq.3a), (5 - B) for A (from eq.1a), and 5 for C in eq.5: 3×B + 5 + 2×B = (5 - B) × 5 which becomes 5×B + 5 = 25 - 5×B In the above equation, subtract 5 from both sides, and add 5×B to both sides: 5×B + 5 - 5 + 5×B = 25 - 5×B - 5 + 5×B which becomes 10×B = 20 Divide both sides by 10: 10×B ÷ 10 = 20 ÷ 10 which means B = 2
Solution
Since B = 2, then: A = 5 - B = 5 - 2 = 3 (from eq.1a) D = 2×B = 2×2 = 4 E = 3×B = 3×2 = 6 F = 5 + 2×B = 5 + 2×2 = 5 + 4 = 9 (from eq.3a) and ABCDEF = 325469