Puzzle for November 11, 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.
Scratchpad
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Hint #1
Add C to both sides of eq.3: C - D + C = E - C + C which becomes eq.3a) 2×C - D = E In eq.4, replace E with 2×C - D (from eq.3a): D + 2×C - D = A + C which becomes 2×C = A + C Subtract C from each side of the above equation: 2×C - C = A + C - C which makes C = A
Hint #2
eq.5 may be written as: F = (B + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (B + C + E) ÷ 3 which becomes 3×F = B + C + E which may be written as eq.5a) 3×F = C + B + E
Hint #3
In eq.5a, replace C with A, and B + E with D + F (from eq.2): 3×F = A + D + F Subtract F from each side of the equation above: 3×F - F = A + D + F - F which becomes eq.5b) 2×F = A + D
Hint #4
In eq.6, substitute 2×F for A + D (from eq.5b): 2×F = B × F Divide both sides of the above equation by F: (2×F) ÷ F = (B × F) ÷ F which makes 2 = B
Hint #5
Substitute 2 for B in eq.1: eq.1a) E = A + 2
Hint #6
Substitute A for C, 2 for B, and A + 2 for E (from eq.1a) in eq.5a: 3×F = A + 2 + A + 2 which becomes eq.5c) 3×F = 2×A + 4
Hint #7
Substitute A for C, and A + 2 for E (from eq.1a) in eq.3a: 2×A - D = A + 2 Subtract 2×A from both sides of the equation above: 2×A - D - 2×A = A + 2 - 2×A which becomes -D = -A + 2 Multiply both sides by (-1): (-1) × (-D) = (-1) × (-A + 2) which becomes eq.3b) D = A - 2
Hint #8
Substitute A - 2 for D (from eq.3b) into eq.5b: 2×F = A + A - 2 which becomes 2×F = 2×A - 2 Add 2 to both sides of the equation above: 2×F + 2 = 2×A - 2 + 2 which becomes eq.5d) 2×F + 2 = 2×A
Hint #9
Substitute 2×F + 2 for 2×A (from eq.5d) in eq.5c: 3×F = 2×F + 2 + 4 which becomes 3×F = 2×F + 6 Subtract 2×F from each side of the above equation: 3×F - 2×F = 2×F + 6 - 2×F which becomes F = 6
Hint #10
Substitute 6 for F in eq.5d: 2×6 + 2 = 2×A which becomes 12 + 2 = 2×A which becomes 14 = 2×A Divide both sides of the above equation by 2: 14 ÷ 2 = 2×A ÷ 2 which makes 7 = A and also makes C = A = 7
Solution
Since A = 7, then: D = A - 2 = 7 - 2 = 5 (from eq.3b) E = A + 2 = 7 + 2 = 9 (from eq.1a) and ABCDEF = 727596