Puzzle for April 26, 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
Add D to both sides of eq.3: B + C + D = D + F + D which becomes B + C + D = 2×D + F In the above equation, replace B + C + D with E + F (from eq.5): E + F = 2×D + F Subtract F from both sides: E + F - F = 2×D + F - F which makes E = 2×D
Hint #2
In eq.6, substitute 2×D for E: D + 2×D = F which makes 3×D = F
Hint #3
Substitute 3×D for F in eq.4: eq.4a) 3×D = A + B
Hint #4
Substitute 3×D for A + B (from eq.4a), 2×D for E, and 3×D for F in eq.1: 3×D + C + D + 2×D + 3×D = 31 which becomes C + 9×D = 31 Subtract 9×D from both sides of the above equation: C + 9×D - 9×D = 31 - 9×D which becomes eq.1a) C = 31 - 9×D
Hint #5
Substitute 3×D for F in eq.3: B + C = D + 3×D which makes eq.3a) B + C = 4×D
Hint #6
Substitute 4×D for B + C (from eq.3a), 2×D for E, and 3×D for F in eq.1: A + 4×D + D + 2×D + 3×D = 31 which becomes A + 10×D = 31 Subtract 10×D from both sides of the above equation: A + 10×D - 10×D = 31 - 10×D which becomes eq.1b) A = 31 - 10×D
Hint #7
Subtract A from both sides of eq.4a: 3×D - A = A + B - A which becomes 3×D - A = B Substitute (31 - 10×D) for A (from eq.1b) in the equation above: 3×D - (31 - 10×D) = B which is equivalent to 3×D - 31 + 10×D = B which becomes eq.4b) 13×D - 31 = B
Solution
Substitute (31 - 10×D) for A (from eq.1b), (31 - 9×D) for C (from eq.1a), 3×D for F, (13×D - 31) for B (from eq.4b), and 2×D for E in eq.2: (31 - 10×D) + (31 - 9×D) + 3×D = (13×D - 31) + 2×D which becomes 62 - 16×D = 15×D - 31 Add 16×D + 31 to both sides of the equation above: 62 - 16×D + 16×D + 31 = 15×D - 31 + 16×D + 31 which simplifies to 93 = 31×D Divide both sides by 31: 93 ÷ 31 = 31×D ÷ 31 which makes 3 = D making A = 31 - 10×D = 31 - 10×3 = 31 - 30 = 1 (from eq.1b) B = 13×D - 31 = 13×3 - 31 = 39 - 31 = 8 (from eq.4b) C = 31 - 9×D = 31 - 9×3 = 31 - 27 = 4 (from eq.1a) E = 2×D = 2×3 = 6 F = 3×D = 3×3 = 9 and ABCDEF = 184369