Puzzle for January 21, 2019 ( )
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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.
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Hint #1
In eq.3, replace E + F with C - D (from eq.4): D + C - D = A which makes C = A
Hint #2
In eq.6, replace C with A: A ÷ A = D which makes 1 = D
Hint #3
Substitute A for C, and 1 for D in eq.2: eq.2a) A + 1 = B
Hint #4
Substitute A + 1 for B (from eq.2a), and 1 for D in eq.5: A + 1 - A = F - 1 which becomes 1 = F - 1 Add 1 to both sides of the equation above: 1 + 1 = F - 1 + 1 which makes 2 = F
Hint #5
Replace D with 1, and F with 2 in eq.3: 1 + E + 2 = A which is equivalent to E + 3 = A Subtract 3 from both sides of the above equation: E + 3 - 3 = A - 3 which means eq.3a) E = A - 3
Solution
Substitute A + 1 for B (from eq.2a), A for C, 1 for D, A - 3 for E (from eq.3a), and 2 for F in eq.1: A + A + 1 + A + 1 + A - 3 + 2 = 29 which simplifies to 4×A + 1 = 29 Subtract 1 from each side of the above equation: 4×A + 1 - 1 = 29 - 1 which becomes 4×A = 28 Divide both sides by 4: 4×A ÷ 4 = 28 ÷ 4 which makes A = 7 making B = A + 1 = 7 + 1 = 8 (from eq.2a) C = A = 7 E = A - 3 = 7 - 3 = 4 (from eq.3a) and ABCDEF = 787142