Puzzle for December 3, 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
In eq.5, replace D + F with B + E (from eq.3), and A with C + F (from eq.2): C + B + E - B = C + F + E - F + B which becomes C + E = C + E + B Subtract C and E from both sides of the above equation: C + E - C - E = C + E + B - C - E which becomes 0 = B
Hint #2
In eq.3, substitute 0 for B: 0 + E = D + F which becomes eq.3a) E = D + F
Hint #3
In eq.4, substitute 0 for B: F - A = A - 0 - E which becomes F - A = A - E Add A and E to both sides of the above equation: F - A + A + E = A - E + A + E which becomes eq.4a) F + E = 2×A
Hint #4
Subtract C from each side of eq.2: A - C = C + F - C which becomes A - C = F Substitute A - C for F in eq.4a: A - C + E = 2×A In the above equation, subtract A from both sides, and add C to both sides: A - C + E - A + C = 2×A - A + C which becomes eq.4b) E = A + C
Hint #5
eq.1 may be written as: B + A + C + D + F + E = 27 In the above equation, substitute 0 for B, E for A + C (from eq.4b), and E for D + F (from eq.3a): 0 + E + E + E = 27 which becomes 3×E = 27 Divide both sides by 3: 3×E ÷ 3 = 27 ÷ 3 which makes E = 9
Hint #6
Substitute 9 for E in eq.4a: F + 9 = 2×A Subtract 9 from each side of the equation above: F + 9 - 9 = 2×A - 9 which becomes eq.4c) F = 2×A - 9
Hint #7
Substitute 9 for E in eq.4b: 9 = A + C Subtract A from each side of the equation above: 9 - A = A + C - A which becomes eq.4d) 9 - A = C
Hint #8
Substitute 9 for E, and 2×A - 9 for F (from eq.4c) in eq.3a: 9 = D + 2×A - 9 In the above equation, subtract 2×A from both sides, and add 9 to both sides: 9 - 2×A + 9 = D + 2×A - 9 - 2×A + 9 which becomes eq.3b) 18 - 2×A = D
Hint #9
Substitute 18 - 2×A for D, 9 for E, and 2×A - 9 for F (from eq.4c) in eq.6: 18 - 2×A = (9 + 2×A - 9) ÷ A which becomes 18 - 2×A = (2×A) ÷ A which makes 18 - 2×A = 2 In the equation above, add 2×A to both sides, and subtract 2 from both sides: 18 - 2×A + 2×A - 2 = 2 + 2×A - 2 which makes 16 = 2×A Divide both sides by 2: 16 ÷ 2 = 2×A ÷ 2 which makes 8 = A
Solution
Since A = 8, then: C = 9 - A = 9 - 8 = 1 (from eq.4d)) D = 18 - 2×A = 18 - 2×8 = 18 - 16 = 2 (from eq.3b) F = 2×A - 9 = 2×8 - 9 = 16 - 9 = 7 (from eq.4c) and ABCDEF = 801297