Puzzle for November 2, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
Scratchpad
Help Area
Hint #1
In eq.3, replace B + E with F (from eq.1): A = F + F which makes A = 2×F
Hint #2
In eq.5, replace A with 2×F: 2×F = D × F Divide both sides of the above equation by F: 2×F ÷ F = (D × F) ÷ F which makes 2 = D
Hint #3
In eq.2, substitute 2 for D: eq.2a) C = 2 + E
Hint #4
Substitute B + E + F for A (from eq.3) in eq.4: B + E + F = B + C + D Subtract B from each side of the equation above: B + E + F - B = B + C + D - B which becomes eq.4a) E + F = C + D
Hint #5
Substitute 2 + E for C (from eq.2a), and 2 for D in eq.4a: E + F = 2 + E + 2 which becomes E + F = 4 + E Subtract E from each side of the above equation: E + F - E = 4 + E - E which makes F = 4 and also makes A = 2×F = 2 × 4 = 8
Hint #6
Substitute 8 for A, 2 + E for C (from eq.2a), and 2 for D in eq.4: 8 = B + 2 + E + 2 which becomes 8 = B + 4 + E Subtract 4 and E from each side of the above equation: 8 - 4 - E = B + 4 + E - 4 - E which becomes eq.4b) 4 - E = B
Hint #7
Substitute 8 for A, 4 - E for B (from eq.4b), and (2 + E) for C (from eq.2a) in eq.6: 8 + E = (4 - E) × (2 + E) which becomes 8 + E = 8 - 2×E + 4×E - E² which becomes 8 + E = 8 + 2×E - E² Subtract 8 and E from both sides of the equation above: 8 + E - 8 - E = 8 + 2×E - E² - 8 - E which becomes eq.6a) 0 = E - E²
Hint #8
eq.6a may be re-written as: 0 = E × (1 - E) In the above equation, either: E = 0 or: 1 - E = 0 which means E = 1 Since E must be a positive integer, then E ≠ 0 and therefore makes E = 1
Hint #9
Substitute 1 for E in eq.4b: 4 - 1 = B which makes 3 = B
Solution
Substitute 1 for E in eq.2a: C = 2 + 1 which makes C = 3 and makes ABCDEF = 833214