Puzzle for November 28, 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
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Hint #1
In eq.3, replace B with A + F (from eq.2): D + F = A + A + F which becomes D + F = 2×A + F Subtract F from both sides of the above equation: D + F - F = 2×A + F - F which makes D = 2×A
Hint #2
In eq.4, replace D with 2×A: 2×A - A = A - E which becomes A = A - E Subtract A from each side of the equation above: A - A = A - E - A which makes 0 = -E which means 0 = E
Hint #3
In eq.5, substitute D + F for A + B (from eq.3), and 0 for E: D + F + C = D + 0 + F Subtract D and F from both sides of the above equation: D + F + C - D - F = D + 0 + F - D - F which simplifies to C = 0
Hint #4
Substitute 0 for C, and 2×A for D in eq.6: B + 0 - F = 2×A ÷ A which becomes B - F = 2 Add F to both sides of the above equation: B - F + F = 2 + F which becomes eq.6a) B = 2 + F
Hint #5
Substitute 2×A for D, and 2 + F for B (from eq.6a) in eq.3: 2×A + F = A + 2 + F Subtract A and F from both sides of the equation above: 2×A + F - A - F = A + 2 + F - A - F which makes A = 2 and also makes D = 2×A = 2 × 2 = 4
Hint #6
Substitute 2 for A, 2 + F for B (from eq.6a), 0 for C and E, and 4 for D in eq.0: 2 + 2 + F + 0 + 4 + 0 + F = 18 which becomes 8 + 2×F = 18 Subtract 8 from both sides of the above equation: 8 + 2×F - 8 = 18 - 8 which makes 2×F = 10 Divide both sides by 2: 2×F ÷ 2 = 10 ÷ 2 which makes F = 5
Solution
Substitute 5 for F in eq.6a: B = 2 + 5 B = 7 and ABCDEF = 270405