Puzzle for June 4, 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
Add E to both sides of eq.4: F - E + E = D + E + E which becomes eq.4a) F = D + 2×E
Hint #2
In eq.1, replace A + B + C with E (from eq.2): E + D + E + F = 18 which becomes D + 2×E + F = 18 In the above equation, replace D + 2×E with F (from eq.4a): F + F = 18 which becomes 2×F = 18 Divide both sides by 2: 2×F ÷ 2 = 18 ÷ 2 which makes F = 9
Hint #3
Add A to each side of eq.5: C + D + E + A = F - A + A which becomes eq.5a) C + D + E + A = F eq.1 may be re-written as: F + B + C + D + E + A = 18 In the above equation, substitute F for C + D + E + A (from eq.5a): F + B + F = 18 which becomes B + 2×F = 18 Substitute 9 for F: B + 2×9 = 18 which becomes B + 18 = 18 Subtract 18 from both sides: B + 18 - 18 = 18 - 18 which means B = 0
Hint #4
Substitute 0 for B in eq.3: D - E = 0 Add E to both sides: D - E + E = 0 + E which means D = E
Hint #5
Substitute D for E, and 9 for F in eq.4a: 9 = D + 2×D which becomes 9 = 3×D Divide both sides by 3: 9 ÷ 3 = 3×D ÷ 3 which makes 3 = D which also makes E = D = 3
Hint #6
Add A and C to both sides of eq.6: A - C + A + C = D - A + A + C which becomes 2×A = D + C which is the same as eq.6a) 2×A = C + D Substitute 2×A for C + D (from eq.6a) in eq.5: 2×A + E = F - A Add A to each side of the above equation: 2×A + E + A = F - A + A which becomes eq.6b) 3×A + E = F
Hint #7
Substitute 3 for E, and 9 for F in eq.6b: 3×A + 3 = 9 Subtract 3 from both sides: 3×A + 3 - 3 = 9 - 3 which means 3×A = 6 Divide both sides by 3: 3×A ÷ 3 = 6 ÷ 3 which makes A = 2
Solution
Substitute 2 for A, and 3 for D in eq.6a: 2×2 = C + 3 which makes 4 = C + 3 Subtract 3 from both sides of the equation above: 4 - 3 = C + 3 - 3 which means 1 = C and ABCDEF = 201339