Puzzle for December 22, 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.
Scratchpad
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Hint #1
Subtract C and A from each side of eq.4: C + F - C - A = A + B - C - A which becomes eq.4a) F - A = B - C Subtract F and C from each side of eq.5: B + F - F - C = A + C + E - F - C which becomes eq.5a) B - C = A + E - F
Hint #2
In eq.5a, replace B - C with F - A (from eq.4a): F - A = A + E - F Add A and F to both sides of the above equation: F - A + A + F = A + E - F + A + F which becomes eq.5b) 2×F = 2×A + E
Hint #3
In eq.5b, substitute (A + D) for F: 2×(A + D) = 2×A + E which becomes 2×A + 2×D = 2×A + E Subtract 2×A from each side of the equation above: 2×A + 2×D - 2×A = 2×A + E - 2×A which makes eq.5c) 2×D = E
Hint #4
In eq.3, replace E with 2×D: A = D + 2×D which makes eq.3a) A = 3×D
Hint #5
In eq.5b, replace A with (3×D), and E with 2×D: 2×F = 2×(3×D) + 2×D which becomes 2×F = 6×D + 2×D which makes 2×F = 8×D Divide both sides of the above equation by 2: 2×F ÷ 2 = 8×D ÷ 2 which makes eq.5d) F = 4×D
Hint #6
eq.6 may be written as: A = (B + C + D + E + F) ÷ 5 Multiply both sides of the above equation by 5: 5 × A = 5 × (B + C + D + E + F) ÷ 5 which becomes eq.6a) 5×A = B + C + D + E + F
Hint #7
In eq.1, substitute 5×A for B + C + D + E + F (from eq.6a): A + 5×A = 36 which makes 6×A = 36 Divide both sides of the above equation by 6: 6×A ÷ 6 = 36 ÷ 6 which makes A = 6
Hint #8
Substitute 6 for A in eq.3a: 6 = 3×D Divide both sides of the above equation by 3: 6 ÷ 3 = 3×D ÷ 3 which makes 2 = D and makes E = 2×D = 2 × 2 = 4 (from eq.5c) F = 4×D = 4 × 2 = 8 (from eq.5d)
Hint #9
Substitute 8 for F, and 6 for A in eq.4: C + 8 = 6 + B Subtract 6 from each side of the equation above: C + 8 - 6 = 6 + B - 6 which makes eq.4b) C + 2 = B
Hint #10
Substitute 6 for A, C + 2 for B (from eq.4b), 2 for D, 4 for E, and 8 for F in eq.6a: 5×6 = C + 2 + C + 2 + 4 + 8 which becomes 30 = 2×C + 16 Subtract 16 from each side of the equation above: 30 - 16 = 2×C + 16 - 16 which makes 14 = 2×C Divide both sides by 2: 14 ÷ 2 = 2×C ÷ 2 which makes 7 = C
Solution
Substitute 7 for C in eq.4b: 7 + 2 = B which makes 9 = B and makes ABCDEF = 697248