Puzzle for July 6, 2019 ( )
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
Help Area
Hint #1
Subtract D from both sides of eq.3: F - D = C + D + E - D which becomes F - D = C + E In the equation above, replace C + E with A + B (from eq.2): eq.3a) F - D = A + B
Hint #2
In eq.1, replace A + B with F - D (from eq.3a), and C + D + E with F (from eq.3): F - D + F + F = 26 which becomes 3×F - D = 26 In the above equation, add D to both sides, and subtract 26 from each side: 3×F - D + D - 26 = 26 + D - 26 which makes eq.1a) 3×F - 26 = D
Hint #3
To make eq.1a true, check several possible values for F and D: If F = 9, then D = 3×9 - 26 = 27 - 26 = 1 If F = 8, then D = 3×8 - 26 = 24 - 26 = -2 If F < 8, then D < -2 Since D must be a non-negative integer, then eq.1a makes F = 9 and D = 1
Hint #4
In eq.3, substitute 9 for F, and 1 for D: 9 = C + 1 + E Subtract 1 from both sides of the above equation: 9 - 1 = C + 1 + E - 1 which makes eq.3a) 8 = C + E
Hint #5
Substitute 1 for D, 9 for F, and 8 for C + E (from eq.3a) in eq.5: 1 + 9 = A + 8 which becomes 10 = A + 8 Subtract 8 from both sides of the equation above: 10 - 8 = A + 8 - 8 which means 2 = A
Hint #6
Substitute 8 for C + E (from eq.3a), and 2 for A in eq.2: 8 = 2 + B Subtract 2 from both sides: 8 - 2 = 2 + B - 2 which makes 6 = B
Hint #7
Substitute 6 for B, 2 for A, and 9 for F in eq.4: 6 + E = 2 + 9 which becomes 6 + E = 11 Subtract 6 from both sides: 6 + E - 6 = 11 - 6 which makes E = 5
Solution
Substitute 5 for E in eq.3a: 8 = C + 5 Subtract 5 from both sides of the above equation: 8 - 5 = C + 5 - 5 which means 3 = C making ABCDEF = 263159