Puzzle for December 20, 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 non-negative integer.
Once again, we thank one of our favorite contributors, Judah S (age 16), for another fun puzzle! Thank you, Judah!
Scratchpad
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Hint #1
In eq.3, replace D - F with A - D + F (from eq.2): F - A = A + A - D + F which becomes F - A = 2×A - D + F In the above equation, subtract F from both sides, and add A and D to both sides: F - A - F + A + D = 2×A - D + F - F + A + D which simplifies to eq.3a) D = 3×A
Hint #2
In eq.6, replace D with 3×A: E = 3×A ÷ A which makes E = 3
Hint #3
In eq.1, replace E with 3: eq.1a) F = A + 3
Hint #4
In eq.3, substitute (A + 3) for F (from eq.1a), and 3×A for D: (A + 3) - A = A + 3×A - (A + 3) which becomes 3 = 4×A - A - 3 which becomes 3 = 3×A - 3 Add 3 to both sides of the above equation: 3 + 3 = 3×A - 3 + 3 which makes 6 = 3×A Divide both sides by 3: 6 ÷ 3 = 3×A ÷ 3 which makes 2 = A
Hint #5
Substitute 2 for A in eq.1a: F = 2 + 3 which makes F = 5
Hint #6
Substitute 2 for A in eq.3a: D = 3×2 which makes D = 6
Hint #7
Substitute 6 for D, 2 for A, and 3 for E in eq.5: 6 - B = 2 × 3 which becomes 6 - B = 6 Subtract 6 from each side of the equation above: 6 - B - 6 = 6 - 6 which makes -B = 0 which means B = 0
Solution
Substitute 2 for A, 6 for D, 0 for B, 3 for E, and 5 for F in eq.4: 2 + C + 6 = 0 + 5 + 3 which becomes 8 + C = 8 Subtract 8 from each side of the above equation: 8 + C - 8 = 8 - 8 which makes C = 0 and makes ABCDEF = 200635